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Suppose we have an ${n \times n}$ matrix ${M}$ that is expressed in block-matrix form as

$\displaystyle M = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$

where ${A}$ is an ${(n-k) \times (n-k)}$ matrix, ${B}$ is an ${(n-k) \times k}$ matrix, ${C}$ is an ${k \times (n-k)}$ matrix, and ${D}$ is a ${k \times k}$ matrix for some ${1 < k < n}$. If ${A}$ is invertible, we can use the technique of Schur complementation to express the inverse of ${M}$ (if it exists) in terms of the inverse of ${A}$, and the other components ${B,C,D}$ of course. Indeed, to solve the equation

$\displaystyle M \begin{pmatrix} x & y \end{pmatrix} = \begin{pmatrix} a & b \end{pmatrix},$

where ${x, a}$ are ${(n-k) \times 1}$ column vectors and ${y,b}$ are ${k \times 1}$ column vectors, we can expand this out as a system

$\displaystyle Ax + By = a$

$\displaystyle Cx + Dy = b.$

Using the invertibility of ${A}$, we can write the first equation as

$\displaystyle x = A^{-1} a - A^{-1} B y \ \ \ \ \ (1)$

and substituting this into the second equation yields

$\displaystyle (D - C A^{-1} B) y = b - C A^{-1} a$

and thus (assuming that ${D - CA^{-1} B}$ is invertible)

$\displaystyle y = - (D - CA^{-1} B)^{-1} CA^{-1} a + (D - CA^{-1} B)^{-1} b$

and then inserting this back into (1) gives

$\displaystyle x = (A^{-1} + A^{-1} B (D - CA^{-1} B)^{-1} C A^{-1}) a - A^{-1} B (D - CA^{-1} B)^{-1} b.$

Comparing this with

$\displaystyle \begin{pmatrix} x & y \end{pmatrix} = M^{-1} \begin{pmatrix} a & b \end{pmatrix},$

we have managed to express the inverse of ${M}$ as

$\displaystyle M^{-1} =$

$\displaystyle \begin{pmatrix} A^{-1} + A^{-1} B (D - CA^{-1} B)^{-1} C A^{-1} & - A^{-1} B (D - CA^{-1} B)^{-1} \\ - (D - CA^{-1} B)^{-1} CA^{-1} & (D - CA^{-1} B)^{-1} \end{pmatrix}. \ \ \ \ \ (2)$

One can consider the inverse problem: given the inverse ${M^{-1}}$ of ${M}$, does one have a nice formula for the inverse ${A^{-1}}$ of the minor ${A}$? Trying to recover this directly from (2) looks somewhat messy. However, one can proceed as follows. Let ${U}$ denote the ${n \times k}$ matrix

$\displaystyle U := \begin{pmatrix} 0 \\ I_k \end{pmatrix}$

(with ${I_k}$ the ${k \times k}$ identity matrix), and let ${V}$ be its transpose:

$\displaystyle V := \begin{pmatrix} 0 & I_k \end{pmatrix}.$

Then for any scalar ${t}$ (which we identify with ${t}$ times the identity matrix), one has

$\displaystyle M + UtV = \begin{pmatrix} A & B \\ C & D+t \end{pmatrix},$

and hence by (2)

$\displaystyle (M+UtV)^{-1} =$

$\displaystyle \begin{pmatrix} A^{-1} + A^{-1} B (D + t - CA^{-1} B)^{-1} C A^{-1} & - A^{-1} B (D + t- CA^{-1} B)^{-1} \\ - (D + t - CA^{-1} B)^{-1} CA^{-1} & (D + t - CA^{-1} B)^{-1} \end{pmatrix}.$

noting that the inverses here will exist for ${t}$ large enough. Taking limits as ${t \rightarrow \infty}$, we conclude that

$\displaystyle \lim_{t \rightarrow \infty} (M+UtV)^{-1} = \begin{pmatrix} A^{-1} & 0 \\ 0 & 0 \end{pmatrix}.$

On the other hand, by the Woodbury matrix identity (discussed in this previous blog post), we have

$\displaystyle (M+UtV)^{-1} = M^{-1} - M^{-1} U (t^{-1} + V M^{-1} U)^{-1} V M^{-1}$

and hence on taking limits and comparing with the preceding identity, one has

$\displaystyle \begin{pmatrix} A^{-1} & 0 \\ 0 & 0 \end{pmatrix} = M^{-1} - M^{-1} U (V M^{-1} U)^{-1} V M^{-1}.$

This achieves the aim of expressing the inverse ${A^{-1}}$ of the minor in terms of the inverse of the full matrix. Taking traces and rearranging, we conclude in particular that

$\displaystyle \mathrm{tr} A^{-1} = \mathrm{tr} M^{-1} - \mathrm{tr} (V M^{-2} U) (V M^{-1} U)^{-1}. \ \ \ \ \ (3)$

In the ${k=1}$ case, this can be simplified to

$\displaystyle \mathrm{tr} A^{-1} = \mathrm{tr} M^{-1} - \frac{e_n^T M^{-2} e_n}{e_n^T M^{-1} e_n} \ \ \ \ \ (4)$

where ${e_n}$ is the ${n^{th}}$ basis column vector.

We can apply this identity to understand how the spectrum of an ${n \times n}$ random matrix ${M}$ relates to that of its top left ${n-1 \times n-1}$ minor ${A}$. Subtracting any complex multiple ${z}$ of the identity from ${M}$ (and hence from ${A}$), we can relate the Stieltjes transform ${s_M(z) := \frac{1}{n} \mathrm{tr}(M-z)^{-1}}$ of ${M}$ with the Stieltjes transform ${s_A(z) := \frac{1}{n-1} \mathrm{tr}(A-z)^{-1}}$ of ${A}$:

$\displaystyle s_A(z) = \frac{n}{n-1} s_M(z) - \frac{1}{n-1} \frac{e_n^T (M-z)^{-2} e_n}{e_n^T (M-z)^{-1} e_n} \ \ \ \ \ (5)$

At this point we begin to proceed informally. Assume for sake of argument that the random matrix ${M}$ is Hermitian, with distribution that is invariant under conjugation by the unitary group ${U(n)}$; for instance, ${M}$ could be drawn from the Gaussian Unitary Ensemble (GUE), or alternatively ${M}$ could be of the form ${M = U D U^*}$ for some real diagonal matrix ${D}$ and ${U}$ a unitary matrix drawn randomly from ${U(n)}$ using Haar measure. To fix normalisations we will assume that the eigenvalues of ${M}$ are typically of size ${O(1)}$. Then ${A}$ is also Hermitian and ${U(n)}$-invariant. Furthermore, the law of ${e_n^T (M-z)^{-1} e_n}$ will be the same as the law of ${u^* (M-z)^{-1} u}$, where ${u}$ is now drawn uniformly from the unit sphere (independently of ${M}$). Diagonalising ${M}$ into eigenvalues ${\lambda_j}$ and eigenvectors ${v_j}$, we have

$\displaystyle u^* (M-z)^{-1} u = \sum_{j=1}^n \frac{|u^* v_j|^2}{\lambda_j - z}.$

One can think of ${u}$ as a random (complex) Gaussian vector, divided by the magnitude of that vector (which, by the Chernoff inequality, will concentrate to ${\sqrt{n}}$). Thus the coefficients ${u^* v_j}$ with respect to the orthonormal basis ${v_1,\dots,v_j}$ can be thought of as independent (complex) Gaussian vectors, divided by that magnitude. Using this and the Chernoff inequality again, we see (for ${z}$ distance ${\sim 1}$ away from the real axis at least) that one has the concentration of measure

$\displaystyle u^* (M-z)^{-1} u \approx \frac{1}{n} \sum_{j=1}^n \frac{1}{\lambda_j - z}$

and thus

$\displaystyle e_n^T (M-z)^{-1} e_n \approx \frac{1}{n} \mathrm{tr} (M-z)^{-1} = s_M(z)$

(that is to say, the diagonal entries of ${(M-z)^{-1}}$ are roughly constant). Similarly we have

$\displaystyle e_n^T (M-z)^{-2} e_n \approx \frac{1}{n} \mathrm{tr} (M-z)^{-2} = \frac{d}{dz} s_M(z).$

Inserting this into (5) and discarding terms of size ${O(1/n^2)}$, we thus conclude the approximate relationship

$\displaystyle s_A(z) \approx s_M(z) + \frac{1}{n} ( s_M(z) - s_M(z)^{-1} \frac{d}{dz} s_M(z) ).$

This can be viewed as a difference equation for the Stieltjes transform of top left minors of ${M}$. Iterating this equation, and formally replacing the difference equation by a differential equation in the large ${n}$ limit, we see that when ${n}$ is large and ${k \approx e^{-t} n}$ for some ${t \geq 0}$, one expects the top left ${k \times k}$ minor ${A_k}$ of ${M}$ to have Stieltjes transform

$\displaystyle s_{A_k}(z) \approx s( t, z ) \ \ \ \ \ (6)$

where ${s(t,z)}$ solves the Burgers-type equation

$\displaystyle \partial_t s(t,z) = s(t,z) - s(t,z)^{-1} \frac{d}{dz} s(t,z) \ \ \ \ \ (7)$

with initial data ${s(0,z) = s_M(z)}$.

Example 1 If ${M}$ is a constant multiple ${M = cI_n}$ of the identity, then ${s_M(z) = \frac{1}{c-z}}$. One checks that ${s(t,z) = \frac{1}{c-z}}$ is a steady state solution to (7), which is unsurprising given that all minors of ${M}$ are also ${c}$ times the identity.

Example 2 If ${M}$ is GUE normalised so that each entry has variance ${\sigma^2/n}$, then by the semi-circular law (see previous notes) one has ${s_M(z) \approx \frac{-z + \sqrt{z^2-4\sigma^2}}{2\sigma^2} = -\frac{2}{z + \sqrt{z^2-4\sigma^2}}}$ (using an appropriate branch of the square root). One can then verify the self-similar solution

$\displaystyle s(t,z) = \frac{-z + \sqrt{z^2 - 4\sigma^2 e^{-t}}}{2\sigma^2 e^{-t}} = -\frac{2}{z + \sqrt{z^2 - 4\sigma^2 e^{-t}}}$

to (7), which is consistent with the fact that a top ${k \times k}$ minor of ${M}$ also has the law of GUE, with each entry having variance ${\sigma^2 / n \approx \sigma^2 e^{-t} / k}$ when ${k \approx e^{-t} n}$.

One can justify the approximation (6) given a sufficiently good well-posedness theory for the equation (7). We will not do so here, but will note that (as with the classical inviscid Burgers equation) the equation can be solved exactly (formally, at least) by the method of characteristics. For any initial position ${z_0}$, we consider the characteristic flow ${t \mapsto z(t)}$ formed by solving the ODE

$\displaystyle \frac{d}{dt} z(t) = s(t,z(t))^{-1} \ \ \ \ \ (8)$

with initial data ${z(0) = z_0}$, ignoring for this discussion the problems of existence and uniqueness. Then from the chain rule, the equation (7) implies that

$\displaystyle \frac{d}{dt} s( t, z(t) ) = s(t,z(t))$

and thus ${s(t,z(t)) = e^t s(0,z_0)}$. Inserting this back into (8) we see that

$\displaystyle z(t) = z_0 + s(0,z_0)^{-1} (1-e^{-t})$

and thus (7) may be solved implicitly via the equation

$\displaystyle s(t, z_0 + s(0,z_0)^{-1} (1-e^{-t}) ) = e^t s(0, z_0) \ \ \ \ \ (9)$

for all ${t}$ and ${z_0}$.

Remark 3 In practice, the equation (9) may stop working when ${z_0 + s(0,z_0)^{-1} (1-e^{-t})}$ crosses the real axis, as (7) does not necessarily hold in this region. It is a cute exercise (ultimately coming from the Cauchy-Schwarz inequality) to show that this crossing always happens, for instance if ${z_0}$ has positive imaginary part then ${z_0 + s(0,z_0)^{-1}}$ necessarily has negative or zero imaginary part.

Example 4 Suppose we have ${s(0,z) = \frac{1}{c-z}}$ as in Example 1. Then (9) becomes

$\displaystyle s( t, z_0 + (c-z_0) (1-e^{-t}) ) = \frac{e^t}{c-z_0}$

for any ${t,z_0}$, which after making the change of variables ${z = z_0 + (c-z_0) (1-e^{-t}) = c - e^{-t} (c - z_0)}$ becomes

$\displaystyle s(t, z ) = \frac{1}{c-z}$

as in Example 1.

Example 5 Suppose we have

$\displaystyle s(0,z) = \frac{-z + \sqrt{z^2-4\sigma^2}}{2\sigma^2} = -\frac{2}{z + \sqrt{z^2-4\sigma^2}}.$

as in Example 2. Then (9) becomes

$\displaystyle s(t, z_0 - \frac{z_0 + \sqrt{z_0^2-4\sigma^2}}{2} (1-e^{-t}) ) = e^t \frac{-z_0 + \sqrt{z_0^2-4\sigma^2}}{2\sigma^2}.$

If we write

$\displaystyle z := z_0 - \frac{z_0 + \sqrt{z_0^2-4\sigma^2}}{2} (1-e^{-t})$

$\displaystyle = \frac{(1+e^{-t}) z_0 - (1-e^{-t}) \sqrt{z_0^2-4\sigma^2}}{2}$

one can calculate that

$\displaystyle z^2 - 4 \sigma^2 e^{-t} = (\frac{(1-e^{-t}) z_0 - (1+e^{-t}) \sqrt{z_0^2-4\sigma^2}}{2})^2$

and hence

$\displaystyle \frac{-z + \sqrt{z^2 - 4\sigma^2 e^{-t}}}{2\sigma^2 e^{-t}} = e^t \frac{-z_0 + \sqrt{z_0^2-4\sigma^2}}{2\sigma^2}$

which gives

$\displaystyle s(t,z) = \frac{-z + \sqrt{z^2 - 4\sigma^2 e^{-t}}}{2\sigma^2 e^{-t}}. \ \ \ \ \ (10)$

One can recover the spectral measure ${\mu}$ from the Stieltjes transform ${s(z)}$ as the weak limit of ${x \mapsto \frac{1}{\pi} \mathrm{Im} s(x+i\varepsilon)}$ as ${\varepsilon \rightarrow 0}$; we write this informally as

$\displaystyle d\mu(x) = \frac{1}{\pi} \mathrm{Im} s(x+i0^+)\ dx.$

In this informal notation, we have for instance that

$\displaystyle \delta_c(x) = \frac{1}{\pi} \mathrm{Im} \frac{1}{c-x-i0^+}\ dx$

which can be interpreted as the fact that the Cauchy distributions ${\frac{1}{\pi} \frac{\varepsilon}{(c-x)^2+\varepsilon^2}}$ converge weakly to the Dirac mass at ${c}$ as ${\varepsilon \rightarrow 0}$. Similarly, the spectral measure associated to (10) is the semicircular measure ${\frac{1}{2\pi \sigma^2 e^{-t}} (4 \sigma^2 e^{-t}-x^2)_+^{1/2}}$.

If we let ${\mu_t}$ be the spectral measure associated to ${s(t,\cdot)}$, then the curve ${e^{-t} \mapsto \mu_t}$ from ${(0,1]}$ to the space of measures is the high-dimensional limit ${n \rightarrow \infty}$ of a Gelfand-Tsetlin pattern (discussed in this previous post), if the pattern is randomly generated amongst all matrices ${M}$ with spectrum asymptotic to ${\mu_0}$ as ${n \rightarrow \infty}$. For instance, if ${\mu_0 = \delta_c}$, then the curve is ${\alpha \mapsto \delta_c}$, corresponding to a pattern that is entirely filled with ${c}$‘s. If instead ${\mu_0 = \frac{1}{2\pi \sigma^2} (4\sigma^2-x^2)_+^{1/2}}$ is a semicircular distribution, then the pattern is

$\displaystyle \alpha \mapsto \frac{1}{2\pi \sigma^2 \alpha} (4\sigma^2 \alpha -x^2)_+^{1/2},$

thus at height ${\alpha}$ from the top, the pattern is semicircular on the interval ${[-2\sigma \sqrt{\alpha}, 2\sigma \sqrt{\alpha}]}$. The interlacing property of Gelfand-Tsetlin patterns translates to the claim that ${\alpha \mu_\alpha(-\infty,\lambda)}$ (resp. ${\alpha \mu_\alpha(\lambda,\infty)}$) is non-decreasing (resp. non-increasing) in ${\alpha}$ for any fixed ${\lambda}$. In principle one should be able to establish these monotonicity claims directly from the PDE (7) or from the implicit solution (9), but it was not clear to me how to do so.

An interesting example of such a limiting Gelfand-Tsetlin pattern occurs when ${\mu_0 = \frac{1}{2} \delta_{-1} + \frac{1}{2} \delta_1}$, which corresponds to ${M}$ being ${2P-I}$, where ${P}$ is an orthogonal projection to a random ${n/2}$-dimensional subspace of ${{\bf C}^n}$. Here we have

$\displaystyle s(0,z) = \frac{1}{2} \frac{1}{-1-z} + \frac{1}{2} \frac{1}{1-z} = \frac{z}{1-z^2}$

and so (9) in this case becomes

$\displaystyle s(t, z_0 + \frac{1-z_0^2}{z_0} (1-e^{-t}) ) = \frac{e^t z_0}{1-z_0^2}$

A tedious calculation then gives the solution

$\displaystyle s(t,z) = \frac{(2e^{-t}-1)z + \sqrt{z^2 - 4e^{-t}(1-e^{-t})}}{2e^{-t}(1-z^2)}. \ \ \ \ \ (11)$

For ${\alpha = e^{-t} > 1/2}$, there are simple poles at ${z=-1,+1}$, and the associated measure is

$\displaystyle \mu_\alpha = \frac{2\alpha-1}{2\alpha} \delta_{-1} + \frac{2\alpha-1}{2\alpha} \delta_1 + \frac{1}{2\pi \alpha(1-x^2)} (4\alpha(1-\alpha)-x^2)_+^{1/2}\ dx.$

This reflects the interlacing property, which forces ${\frac{2\alpha-1}{2\alpha} \alpha n}$ of the ${\alpha n}$ eigenvalues of the ${\alpha n \times \alpha n}$ minor to be equal to ${-1}$ (resp. ${+1}$). For ${\alpha = e^{-t} \leq 1/2}$, the poles disappear and one just has

$\displaystyle \mu_\alpha = \frac{1}{2\pi \alpha(1-x^2)} (4\alpha(1-\alpha)-x^2)_+^{1/2}\ dx.$

For ${\alpha=1/2}$, one has an inverse semicircle distribution

$\displaystyle \mu_{1/2} = \frac{1}{\pi} (1-x^2)_+^{-1/2}.$

There is presumably a direct geometric explanation of this fact (basically describing the singular values of the product of two random orthogonal projections to half-dimensional subspaces of ${{\bf C}^n}$), but I do not know of one off-hand.

The evolution of ${s(t,z)}$ can also be understood using the ${R}$-transform and ${S}$-transform from free probability. Formally, letlet ${z(t,s)}$ be the inverse of ${s(t,z)}$, thus

$\displaystyle s(t,z(t,s)) = s$

for all ${t,s}$, and then define the ${R}$-transform

$\displaystyle R(t,s) := z(t,-s) - \frac{1}{s}.$

The equation (9) may be rewritten as

$\displaystyle z( t, e^t s ) = z(0,s) + s^{-1} (1-e^{-t})$

and hence

$\displaystyle R(t, -e^t s) = R(0, -s)$

or equivalently

$\displaystyle R(t,s) = R(0, e^{-t} s). \ \ \ \ \ (12)$

See these previous notes for a discussion of free probability topics such as the ${R}$-transform.

Example 6 If ${s(t,z) = \frac{1}{c-z}}$ then the ${R}$ transform is ${R(t,s) = c}$.

Example 7 If ${s(t,z)}$ is given by (10), then the ${R}$ transform is

$\displaystyle R(t,s) = \sigma^2 e^{-t} s.$

Example 8 If ${s(t,z)}$ is given by (11), then the ${R}$ transform is

$\displaystyle R(t,s) = \frac{-1 + \sqrt{1 + 4 s^2 e^{-2t}}}{2 s e^{-t}}.$

This simple relationship (12) is essentially due to Nica and Speicher (thanks to Dima Shylakhtenko for this reference). It has the remarkable consequence that when ${\alpha = 1/m}$ is the reciprocal of a natural number ${m}$, then ${\mu_{1/m}}$ is the free arithmetic mean of ${m}$ copies of ${\mu}$, that is to say ${\mu_{1/m}}$ is the free convolution ${\mu \boxplus \dots \boxplus \mu}$ of ${m}$ copies of ${\mu}$, pushed forward by the map ${\lambda \rightarrow \lambda/m}$. In terms of random matrices, this is asserting that the top ${n/m \times n/m}$ minor of a random matrix ${M}$ has spectral measure approximately equal to that of an arithmetic mean ${\frac{1}{m} (M_1 + \dots + M_m)}$ of ${m}$ independent copies of ${M}$, so that the process of taking top left minors is in some sense a continuous analogue of the process of taking freely independent arithmetic means. There ought to be a geometric proof of this assertion, but I do not know of one. In the limit ${m \rightarrow \infty}$ (or ${\alpha \rightarrow 0}$), the ${R}$-transform becomes linear and the spectral measure becomes semicircular, which is of course consistent with the free central limit theorem.

In a similar vein, if one defines the function

$\displaystyle \omega(t,z) := \alpha \int_{\bf R} \frac{zx}{1-zx}\ d\mu_\alpha(x) = e^{-t} (- 1 - z^{-1} s(t, z^{-1}))$

and inverts it to obtain a function ${z(t,\omega)}$ with

$\displaystyle \omega(t, z(t,\omega)) = \omega$

for all ${t, \omega}$, then the ${S}$-transform ${S(t,\omega)}$ is defined by

$\displaystyle S(t,\omega) := \frac{1+\omega}{\omega} z(t,\omega).$

Writing

$\displaystyle s(t,z) = - z^{-1} ( 1 + e^t \omega(t, z^{-1}) )$

for any ${t}$, ${z}$, we have

$\displaystyle z_0 + s(0,z_0)^{-1} (1-e^{-t}) = z_0 \frac{\omega(0,z_0^{-1})+e^{-t}}{\omega(0,z_0^{-1})+1}$

and so (9) becomes

$\displaystyle - z_0^{-1} \frac{\omega(0,z_0^{-1})+1}{\omega(0,z_0^{-1})+e^{-t}} (1 + e^{t} \omega(t, z_0^{-1} \frac{\omega(0,z_0^{-1})+1}{\omega(0,z_0^{-1})+e^{-t}}))$

$\displaystyle = - e^t z_0^{-1} (1 + \omega(0, z_0^{-1}))$

which simplifies to

$\displaystyle \omega(t, z_0^{-1} \frac{\omega(0,z_0^{-1})+1}{\omega(0,z_0^{-1})+e^{-t}})) = \omega(0, z_0^{-1});$

replacing ${z_0}$ by ${z(0,\omega)^{-1}}$ we obtain

$\displaystyle \omega(t, z(0,\omega) \frac{\omega+1}{\omega+e^{-t}}) = \omega$

and thus

$\displaystyle z(0,\omega)\frac{\omega+1}{\omega+e^{-t}} = z(t, \omega)$

and hence

$\displaystyle S(0, \omega) = \frac{\omega+e^{-t}}{\omega+1} S(t, \omega).$

One can compute ${\frac{\omega+e^{-t}}{\omega+1}}$ to be the ${S}$-transform of the measure ${(1-\alpha) \delta_0 + \alpha \delta_1}$; from the link between ${S}$-transforms and free products (see e.g. these notes of Guionnet), we conclude that ${(1-\alpha)\delta_0 + \alpha \mu_\alpha}$ is the free product of ${\mu_1}$ and ${(1-\alpha) \delta_0 + \alpha \delta_1}$. This is consistent with the random matrix theory interpretation, since ${(1-\alpha)\delta_0 + \alpha \mu_\alpha}$ is also the spectral measure of ${PMP}$, where ${P}$ is the orthogonal projection to the span of the first ${\alpha n}$ basis elements, so in particular ${P}$ has spectral measure ${(1-\alpha) \delta_0 + \alpha \delta_1}$. If ${M}$ is unitarily invariant then (by a fundamental result of Voiculescu) it is asymptotically freely independent of ${P}$, so the spectral measure of ${PMP = P^{1/2} M P^{1/2}}$ is asymptotically the free product of that of ${M}$ and of ${P}$.

Szemerédi’s theorem asserts that all subsets of the natural numbers of positive density contain arbitrarily long arithmetic progressions.  Roth’s theorem is the special case when one considers arithmetic progressions of length three.  Both theorems have many important proofs using tools from additive combinatorics, (higher order) Fourier analysis, (hyper) graph regularity theory, and ergodic theory.  However, the original proof by Endre Szemerédi, while extremely intricate, was purely combinatorial (and in particular “elementary”) and almost entirely self-contained, except for an invocation of the van der Waerden theorem.  It is also notable for introducing a prototype of what is now known as the Szemerédi regularity lemma.

Back in 2005, I rewrote Szemerédi’s original proof in order to understand it better, however my rewrite ended up being about the same length as the original argument and was probably only usable to myself.  In 2012, after Szemerédi was awarded the Abel prize, I revisited this argument with the intention to try to write up a more readable version of the proof, but ended up just presenting some ingredients of the argument in a blog post, rather than try to rewrite the whole thing.  In that post, I suspected that the cleanest way to write up the argument would be through the language of nonstandard analysis (perhaps in an iterated hyperextension that could handle various hierarchies of infinitesimals), but was unable to actually achieve any substantial simplifications by passing to the nonstandard world.

A few weeks ago, I participated in a week-long workshop at the American Institute of Mathematics on “Nonstandard methods in combinatorial number theory”, and spent some time in a working group with Shabnam Akhtari, Irfam Alam, Renling Jin, Steven Leth, Karl Mahlburg, Paul Potgieter, and Henry Towsner to try to obtain a manageable nonstandard version of Szemerédi’s original proof.  We didn’t end up being able to do so – in fact there are now signs that perhaps nonstandard analysis is not the optimal framework in which to place this argument – but we did at least clarify the existing standard argument, to the point that I was able to go back to my original rewrite of the proof and present it in a more civilised form, which I am now uploading here as an unpublished preprint.   There are now a number of simplifications to the proof.  Firstly, one no longer needs the full strength of the regularity lemma; only the simpler “weak” regularity lemma of Frieze and Kannan is required.  Secondly, the proof has been “factored” into a number of stand-alone propositions of independent interest, in particular involving just (families of) one-dimensional arithmetic progressions rather than the complicated-looking multidimensional arithmetic progressions that occur so frequently in the original argument of Szemerédi.  Finally, the delicate manipulations of densities and epsilons via double counting arguments in Szemerédi’s original paper have been abstracted into a certain key property of families of arithmetic progressions that I call the “double counting property”.

The factoring mentioned above is particularly simple in the case of proving Roth’s theorem, which is now presented separately in the above writeup.  Roth’s theorem seeks to locate a length three progression ${(P(1),P(2),P(3)) = (a, a+r, a+2r)}$ in which all three elements lie in a single set.  This will be deduced from an easier variant of the theorem in which one locates (a family of) length three progressions in which just the first two elements ${P(1), P(2)}$ of the progression lie in a good set (and some other properties of the family are also required).  This is in turn derived from an even easier variant in which now just the first element of the progression is required to be in the good set.

More specifically, Roth’s theorem is now deduced from

Theorem 1.5.  Let ${L}$ be a natural number, and let ${S}$ be a set of integers of upper density at least ${1-1/10L}$.  Then, whenever ${S}$ is partitioned into finitely many colour classes, there exists a colour class ${A}$ and a family ${(P_l(1),P_l(2),P_l(3))_{l=1}^L}$ of 3-term arithmetic progressions with the following properties:

1. For each ${l}$, ${P_l(1)}$ and ${P_l(2)}$ lie in ${A}$.
2. For each ${l}$, ${P_l(3)}$ lie in ${S}$.
3. The ${P_l(3)}$ for ${l=1,\dots,L}$ are in arithmetic progression.

The situation in this theorem is depicted by the following diagram, in which elements of $A$ are in blue and elements of $S$ are in grey:

Theorem 1.5 is deduced in turn from the following easier variant:

Theorem 1.6.  Let ${L}$ be a natural number, and let ${S}$ be a set of integers of upper density at least ${1-1/10L}$.  Then, whenever ${S}$ is partitioned into finitely many colour classes, there exists a colour class ${A}$ and a family ${(P_l(1),P_l(2),P_l(3))_{l=1}^L}$ of 3-term arithmetic progressions with the following properties:

1. For each ${l}$, ${P_l(1)}$ lie in ${A}$.
2. For each ${l}$, ${P_l(2)}$ and ${P_l(3)}$ lie in ${S}$.
3. The ${P_l(2)}$ for ${l=1,\dots,L}$ are in arithmetic progression.

The situation here is described by the figure below.

Theorem 1.6 is easy to prove.  To derive Theorem 1.5 from Theorem 1.6, or to derive Roth’s theorem from Theorem 1.5, one uses double counting arguments, van der Waerden’s theorem, and the weak regularity lemma, largely as described in this previous blog post; see the writeup for the full details.  (I would be interested in seeing a shorter proof of Theorem 1.5 though that did not go through these arguments, and did not use the more powerful theorems of  Roth or Szemerédi.)

Fix a non-negative integer ${k}$. Define an (weak) integer partition of length ${k}$ to be a tuple ${\lambda = (\lambda_1,\dots,\lambda_k)}$ of non-increasing non-negative integers ${\lambda_1 \geq \dots \geq \lambda_k \geq 0}$. (Here our partitions are “weak” in the sense that we allow some parts of the partition to be zero. Henceforth we will omit the modifier “weak”, as we will not need to consider the more usual notion of “strong” partitions.) To each such partition ${\lambda}$, one can associate a Young diagram consisting of ${k}$ left-justified rows of boxes, with the ${i^{th}}$ row containing ${\lambda_i}$ boxes. A semi-standard Young tableau (or Young tableau for short) ${T}$ of shape ${\lambda}$ is a filling of these boxes by integers in ${\{1,\dots,k\}}$ that is weakly increasing along rows (moving rightwards) and strictly increasing along columns (moving downwards). The collection of such tableaux will be denoted ${{\mathcal T}_\lambda}$. The weight ${|T|}$ of a tableau ${T}$ is the tuple ${(n_1,\dots,n_k)}$, where ${n_i}$ is the number of occurrences of the integer ${i}$ in the tableau. For instance, if ${k=3}$ and ${\lambda = (6,4,2)}$, an example of a Young tableau of shape ${\lambda}$ would be

$\displaystyle \begin{tabular}{|c|c|c|c|c|c|} \hline 1 & 1 & 1 & 2 & 3 & 3 \\ \cline{1-6} 2 & 2 & 2 &3\\ \cline{1-4} 3 & 3\\ \cline{1-2} \end{tabular}$

The weight here would be ${|T| = (3,4,5)}$.

To each partition ${\lambda}$ one can associate the Schur polynomial ${s_\lambda(u_1,\dots,u_k)}$ on ${k}$ variables ${u = (u_1,\dots,u_k)}$, which we will define as

$\displaystyle s_\lambda(u) := \sum_{T \in {\mathcal T}_\lambda} u^{|T|}$

using the multinomial convention

$\displaystyle (u_1,\dots,u_k)^{(n_1,\dots,n_k)} := u_1^{n_1} \dots u_k^{n_k}.$

Thus for instance the Young tableau ${T}$ given above would contribute a term ${u_1^3 u_2^4 u_3^5}$ to the Schur polynomial ${s_{(6,4,2)}(u_1,u_2,u_3)}$. In the case of partitions of the form ${(n,0,\dots,0)}$, the Schur polynomial ${s_{(n,0,\dots,0)}}$ is just the complete homogeneous symmetric polynomial ${h_n}$ of degree ${n}$ on ${k}$ variables:

$\displaystyle s_{(n,0,\dots,0)}(u_1,\dots,u_k) := \sum_{n_1,\dots,n_k \geq 0: n_1+\dots+n_k = n} u_1^{n_1} \dots u_k^{n_k},$

thus for instance

$\displaystyle s_{(3,0)}(u_1,u_2) = u_1^3 + u_1^2 u_2 + u_1 u_2^2 + u_2^3.$

Schur polyomials are ubiquitous in the algebraic combinatorics of “type ${A}$ objects” such as the symmetric group ${S_k}$, the general linear group ${GL_k}$, or the unitary group ${U_k}$. For instance, one can view ${s_\lambda}$ as the character of an irreducible polynomial representation of ${GL_k({\bf C})}$ associated with the partition ${\lambda}$. However, we will not focus on these interpretations of Schur polynomials in this post.

This definition of Schur polynomials allows for a way to describe the polynomials recursively. If ${k > 1}$ and ${T}$ is a Young tableau of shape ${\lambda = (\lambda_1,\dots,\lambda_k)}$, taking values in ${\{1,\dots,k\}}$, one can form a sub-tableau ${T'}$ of some shape ${\lambda' = (\lambda'_1,\dots,\lambda'_{k-1})}$ by removing all the appearances of ${k}$ (which, among other things, necessarily deletes the ${k^{th}}$ row). For instance, with ${T}$ as in the previous example, the sub-tableau ${T'}$ would be

$\displaystyle \begin{tabular}{|c|c|c|c|} \hline 1 & 1 & 1 & 2 \\ \cline{1-4} 2 & 2 & 2 \\ \cline{1-3} \end{tabular}$

and the reduced partition ${\lambda'}$ in this case is ${(4,3)}$. As Young tableaux are required to be strictly increasing down columns, we can see that the reduced partition ${\lambda'}$ must intersperse the original partition ${\lambda}$ in the sense that

$\displaystyle \lambda_{i+1} \leq \lambda'_i \leq \lambda_i \ \ \ \ \ (1)$

for all ${1 \leq i \leq k-1}$; we denote this interspersion relation as ${\lambda' \prec \lambda}$ (though we caution that this is not intended to be a partial ordering). In the converse direction, if ${\lambda' \prec \lambda}$ and ${T'}$ is a Young tableau with shape ${\lambda'}$ with entries in ${\{1,\dots,k-1\}}$, one can form a Young tableau ${T}$ with shape ${\lambda}$ and entries in ${\{1,\dots,k\}}$ by appending to ${T'}$ an entry of ${k}$ in all the boxes that appear in the ${\lambda}$ shape but not the ${\lambda'}$ shape. This one-to-one correspondence leads to the recursion

$\displaystyle s_\lambda(u) = \sum_{\lambda' \prec \lambda} s_{\lambda'}(u') u_k^{|\lambda| - |\lambda'|} \ \ \ \ \ (2)$

where ${u = (u_1,\dots,u_k)}$, ${u' = (u_1,\dots,u_{k-1})}$, and the size ${|\lambda|}$ of a partition ${\lambda = (\lambda_1,\dots,\lambda_k)}$ is defined as ${|\lambda| := \lambda_1 + \dots + \lambda_k}$.

One can use this recursion (2) to prove some further standard identities for Schur polynomials, such as the determinant identity

$\displaystyle s_\lambda(u) V(u) = \det( u_i^{\lambda_j+k-j} )_{1 \leq i,j \leq k} \ \ \ \ \ (3)$

for ${u=(u_1,\dots,u_k)}$, where ${V(u)}$ denotes the Vandermonde determinant

$\displaystyle V(u) := \prod_{1 \leq i < j \leq k} (u_i - u_j), \ \ \ \ \ (4)$

or the Jacobi-Trudi identity

$\displaystyle s_\lambda(u) = \det( h_{\lambda_j - j + i}(u) )_{1 \leq i,j \leq k}, \ \ \ \ \ (5)$

with the convention that ${h_d(u) = 0}$ if ${d}$ is negative. Thus for instance

$\displaystyle s_{(1,1,0,\dots,0)}(u) = h_1^2(u) - h_0(u) h_2(u) = \sum_{1 \leq i < j \leq k} u_i u_j.$

We review the (standard) derivation of these identities via (2) below the fold. Among other things, these identities show that the Schur polynomials are symmetric, which is not immediately obvious from their definition.

One can also iterate (2) to write

$\displaystyle s_\lambda(u) = \sum_{() = \lambda^0 \prec \lambda^1 \prec \dots \prec \lambda^k = \lambda} \prod_{j=1}^k u_j^{|\lambda^j| - |\lambda^{j-1}|} \ \ \ \ \ (6)$

where the sum is over all tuples ${\lambda^1,\dots,\lambda^k}$, where each ${\lambda^j}$ is a partition of length ${j}$ that intersperses the next partition ${\lambda^{j+1}}$, with ${\lambda^k}$ set equal to ${\lambda}$. We will call such a tuple an integral Gelfand-Tsetlin pattern based at ${\lambda}$.

One can generalise (6) by introducing the skew Schur functions

$\displaystyle s_{\lambda/\mu}(u) := \sum_{\mu = \lambda^i \prec \dots \prec \lambda^k = \lambda} \prod_{j=i+1}^k u_j^{|\lambda^j| - |\lambda^{j-1}|} \ \ \ \ \ (7)$

for ${u = (u_{i+1},\dots,u_k)}$, whenever ${\lambda}$ is a partition of length ${k}$ and ${\mu}$ a partition of length ${i}$ for some ${0 \leq i \leq k}$, thus the Schur polynomial ${s_\lambda}$ is also the skew Schur polynomial ${s_{\lambda /()}}$ with ${i=0}$. (One could relabel the variables here to be something like ${(u_1,\dots,u_{k-i})}$ instead, but this labeling seems slightly more natural, particularly in view of identities such as (8) below.)

By construction, we have the decomposition

$\displaystyle s_{\lambda/\nu}(u_{i+1},\dots,u_k) = \sum_\mu s_{\mu/\nu}(u_{i+1},\dots,u_j) s_{\lambda/\mu}(u_{j+1},\dots,u_k) \ \ \ \ \ (8)$

whenever ${0 \leq i \leq j \leq k}$, and ${\nu, \mu, \lambda}$ are partitions of lengths ${i,j,k}$ respectively. This gives another recursive way to understand Schur polynomials and skew Schur polynomials. For instance, one can use it to establish the generalised Jacobi-Trudi identity

$\displaystyle s_{\lambda/\mu}(u) = \det( h_{\lambda_j - j - \mu_i + i}(u) )_{1 \leq i,j \leq k}, \ \ \ \ \ (9)$

with the convention that ${\mu_i = 0}$ for ${i}$ larger than the length of ${\mu}$; we do this below the fold.

The Schur polynomials (and skew Schur polynomials) are “discretised” (or “quantised”) in the sense that their parameters ${\lambda, \mu}$ are required to be integer-valued, and their definition similarly involves summation over a discrete set. It turns out that there are “continuous” (or “classical”) analogues of these functions, in which the parameters ${\lambda,\mu}$ now take real values rather than integers, and are defined via integration rather than summation. One can view these continuous analogues as a “semiclassical limit” of their discrete counterparts, in a manner that can be made precise using the machinery of geometric quantisation, but we will not do so here.

The continuous analogues can be defined as follows. Define a real partition of length ${k}$ to be a tuple ${\lambda = (\lambda_1,\dots,\lambda_k)}$ where ${\lambda_1 \geq \dots \geq \lambda_k \geq 0}$ are now real numbers. We can define the relation ${\lambda' \prec \lambda}$ of interspersion between a length ${k-1}$ real partition ${\lambda' = (\lambda'_1,\dots,\lambda'_{k-1})}$ and a length ${k}$ real partition ${\lambda = (\lambda_1,\dots,\lambda_{k})}$ precisely as before, by requiring that the inequalities (1) hold for all ${1 \leq i \leq k-1}$. We can then define the continuous Schur functions ${S_\lambda(x)}$ for ${x = (x_1,\dots,x_k) \in {\bf R}^k}$ recursively by defining

$\displaystyle S_{()}() = 1$

and

$\displaystyle S_\lambda(x) = \int_{\lambda' \prec \lambda} S_{\lambda'}(x') \exp( (|\lambda| - |\lambda'|) x_k ) \ \ \ \ \ (10)$

for ${k \geq 1}$ and ${\lambda}$ of length ${k}$, where ${x' := (x_1,\dots,x_{k-1})}$ and the integral is with respect to ${k-1}$-dimensional Lebesgue measure, and ${|\lambda| = \lambda_1 + \dots + \lambda_k}$ as before. Thus for instance

$\displaystyle S_{(\lambda_1)}(x_1) = \exp( \lambda_1 x_1 )$

and

$\displaystyle S_{(\lambda_1,\lambda_2)}(x_1,x_2) = \int_{\lambda_2}^{\lambda_1} \exp( \lambda'_1 x_1 + (\lambda_1+\lambda_2-\lambda'_1) x_2 )\ d\lambda'_1.$

More generally, we can define the continuous skew Schur functions ${S_{\lambda/\mu}(x)}$ for ${\lambda}$ of length ${k}$, ${\mu}$ of length ${j \leq k}$, and ${x = (x_{j+1},\dots,x_k) \in {\bf R}^{k-j}}$ recursively by defining

$\displaystyle S_{\mu/\mu}() = 1$

and

$\displaystyle S_{\lambda/\mu}(x) = \int_{\lambda' \prec \lambda} S_{\lambda'/\mu}(x') \exp( (|\lambda| - |\lambda'|) x_k )$

for ${k > j}$. Thus for instance

$\displaystyle S_{(\lambda_1,\lambda_2,\lambda_3)/(\mu_1,\mu_2)}(x_3) = 1_{\lambda_3 \leq \mu_2 \leq \lambda_2 \leq \mu_1 \leq \lambda_1} \exp( x_3 (\lambda_1+\lambda_2+\lambda_3 - \mu_1 - \mu_2 ))$

and

$\displaystyle S_{(\lambda_1,\lambda_2,\lambda_3)/(\mu_1)}(x_2, x_3) = \int_{\lambda_2 \leq \lambda'_2 \leq \lambda_2, \mu_1} \int_{\mu_1, \lambda_2 \leq \lambda'_1 \leq \lambda_1}$

$\displaystyle \exp( x_2 (\lambda'_1+\lambda'_2 - \mu_1) + x_3 (\lambda_1+\lambda_2+\lambda_3 - \lambda'_1 - \lambda'_2))\ d\lambda'_1 d\lambda'_2.$

By expanding out the recursion, one obtains the analogue

$\displaystyle S_\lambda(x) = \int_{\lambda^1 \prec \dots \prec \lambda^k = \lambda} \exp( \sum_{j=1}^k x_j (|\lambda^j| - |\lambda^{j-1}|))\ d\lambda^1 \dots d\lambda^{k-1},$

of (6), and more generally one has

$\displaystyle S_{\lambda/\mu}(x) = \int_{\mu = \lambda^i \prec \dots \prec \lambda^k = \lambda} \exp( \sum_{j=i+1}^k x_j (|\lambda^j| - |\lambda^{j-1}|))\ d\lambda^{i+1} \dots d\lambda^{k-1}.$

We will call the tuples ${(\lambda^1,\dots,\lambda^k)}$ in the first integral real Gelfand-Tsetlin patterns based at ${\lambda}$. The analogue of (8) is then

$\displaystyle S_{\lambda/\nu}(x_{i+1},\dots,x_k) = \int S_{\mu/\nu}(x_{i+1},\dots,x_j) S_{\lambda/\mu}(x_{j+1},\dots,x_k)\ d\mu$

where the integral is over all real partitions ${\mu}$ of length ${j}$, with Lebesgue measure.

By approximating various integrals by their Riemann sums, one can relate the continuous Schur functions to their discrete counterparts by the limiting formula

$\displaystyle N^{-k(k-1)/2} s_{\lfloor N \lambda \rfloor}( \exp[ x/N ] ) \rightarrow S_\lambda(x) \ \ \ \ \ (11)$

as ${N \rightarrow \infty}$ for any length ${k}$ real partition ${\lambda = (\lambda_1,\dots,\lambda_k)}$ and any ${x = (x_1,\dots,x_k) \in {\bf R}^k}$, where

$\displaystyle \lfloor N \lambda \rfloor := ( \lfloor N \lambda_1 \rfloor, \dots, \lfloor N \lambda_k \rfloor )$

and

$\displaystyle \exp[x/N] := (\exp(x_1/N), \dots, \exp(x_k/N)).$

More generally, one has

$\displaystyle N^{j(j-1)/2-k(k-1)/2} s_{\lfloor N \lambda \rfloor / \lfloor N \mu \rfloor}( \exp[ x/N ] ) \rightarrow S_{\lambda/\mu}(x)$

as ${N \rightarrow \infty}$ for any length ${k}$ real partition ${\lambda}$, any length ${j}$ real partition ${\mu}$ with ${0 \leq j \leq k}$, and any ${x = (x_{j+1},\dots,x_k) \in {\bf R}^{k-j}}$.

As a consequence of these limiting formulae, one expects all of the discrete identities above to have continuous counterparts. This is indeed the case; below the fold we shall prove the discrete and continuous identities in parallel. These are not new results by any means, but I was not able to locate a good place in the literature where they are explicitly written down, so I thought I would try to do so here (primarily for my own internal reference, but perhaps the calculations will be worthwhile to some others also).

The determinant ${\det_n(A)}$ of an ${n \times n}$ matrix (with coefficients in an arbitrary field) obey many useful identities, starting of course with the fundamental multiplicativity ${\det_n(AB) = \det_n(A) \det_n(B)}$ for ${n \times n}$ matrices ${A,B}$. This multiplicativity can in turn be used to establish many further identities; in particular, as shown in this previous post, it implies the Schur determinant identity

$\displaystyle \det_{n+k}\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \det_n(A) \det_k( D - C A^{-1} B ) \ \ \ \ \ (1)$

whenever ${A}$ is an invertible ${n \times n}$ matrix, ${B}$ is an ${n \times k}$ matrix, ${C}$ is a ${k \times n}$ matrix, and ${D}$ is a ${k \times k}$ matrix. The matrix ${D - CA^{-1} B}$ is known as the Schur complement of the block ${A}$.

I only recently discovered that this identity in turn immediately implies what I always found to be a somewhat curious identity, namely the Dodgson condensation identity (also known as the Desnanot-Jacobi identity)

$\displaystyle \det_n(M) \det_{n-2}(M^{1,n}_{1,n}) = \det_{n-1}( M^1_1 ) \det_{n-1}(M^n_n)$

$\displaystyle - \det_{n-1}(M^1_n) \det_{n-1}(M^n_1)$

for any ${n \geq 3}$ and ${n \times n}$ matrix ${M}$, where ${M^i_j}$ denotes the ${n-1 \times n-1}$ matrix formed from ${M}$ by removing the ${i^{th}}$ row and ${j^{th}}$ column, and similarly ${M^{i,i'}_{j,j'}}$ denotes the ${n-2 \times n-2}$ matrix formed from ${M}$ by removing the ${i^{th}}$ and ${(i')^{th}}$ rows and ${j^{th}}$ and ${(j')^{th}}$ columns. Thus for instance when ${n=3}$ we obtain

$\displaystyle \det_3 \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \cdot e$

$\displaystyle = \det_2 \begin{pmatrix} e & f \\ h & i \end{pmatrix} \cdot \det_2 \begin{pmatrix} a & b \\ d & e \end{pmatrix}$

$\displaystyle - \det_2 \begin{pmatrix} b & c \\ e & f \end{pmatrix} \cdot \det_2 \begin{pmatrix} d & e \\ g & h \end{pmatrix}$

for any scalars ${a,b,c,d,e,f,g,h,i}$. (Charles Dodgson, better known by his pen name Lewis Caroll, is of course also known for writing “Alice in Wonderland” and “Through the Looking Glass“.)

The derivation is not new; it is for instance noted explicitly in this paper of Brualdi and Schneider, though I do not know if this is the earliest place in the literature where it can be found. (EDIT: Apoorva Khare has pointed out to me that the original arguments of Dodgson can be interpreted as implicitly following this derivation.) I thought it is worth presenting the short derivation here, though.

Firstly, by swapping the first and ${(n-1)^{th}}$ rows, and similarly for the columns, it is easy to see that the Dodgson condensation identity is equivalent to the variant

$\displaystyle \det_n(M) \det_{n-2}(M^{n-1,n}_{n-1,n}) = \det_{n-1}( M^{n-1}_{n-1} ) \det_{n-1}(M^n_n) \ \ \ \ \ (2)$

$\displaystyle - \det_{n-1}(M^{n-1}_n) \det_{n-1}(M^n_{n-1}).$

Now write

$\displaystyle M = \begin{pmatrix} A & B_1 & B_2 \\ C_1 & d_{11} & d_{12} \\ C_2 & d_{21} & d_{22} \end{pmatrix}$

where ${A}$ is an ${n-2 \times n-2}$ matrix, ${B_1, B_2}$ are ${n-2 \times 1}$ column vectors, ${C_1, C_2}$ are ${1 \times n-2}$ row vectors, and ${d_{11}, d_{12}, d_{21}, d_{22}}$ are scalars. If ${A}$ is invertible, we may apply the Schur determinant identity repeatedly to conclude that

$\displaystyle \det_n(M) = \det_{n-2}(A) \det_2 \begin{pmatrix} d_{11} - C_1 A^{-1} B_1 & d_{12} - C_1 A^{-1} B_2 \\ d_{21} - C_2 A^{-1} B_1 & d_{22} - C_2 A^{-1} B_2 \end{pmatrix}$

$\displaystyle \det_{n-2} (M^{n-1,n}_{n-1,n}) = \det_{n-2}(A)$

$\displaystyle \det_{n-1}( M^{n-1}_{n-1} ) = \det_{n-2}(A) (d_{22} - C_2 A^{-1} B_2 )$

$\displaystyle \det_{n-1}( M^{n-1}_{n} ) = \det_{n-2}(A) (d_{21} - C_2 A^{-1} B_1 )$

$\displaystyle \det_{n-1}( M^{n}_{n-1} ) = \det_{n-2}(A) (d_{12} - C_1 A^{-1} B_2 )$

$\displaystyle \det_{n-1}( M^{n}_{n} ) = \det_{n-2}(A) (d_{11} - C_1 A^{-1} B_1 )$

and the claim (2) then follows by a brief calculation (and the explicit form ${\det_2 \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad-bc}$ of the ${2 \times 2}$ determinant). To remove the requirement that ${A}$ be invertible, one can use a limiting argument, noting that one can work without loss of generality in an algebraically closed field, and in such a field, the set of invertible matrices is dense in the Zariski topology. (In the case when the scalars are reals or complexes, one can just use density in the ordinary topology instead if desired.)

The same argument gives the more general determinant identity of Sylvester

$\displaystyle \det_n(M) \det_{n-k}(M^S_S)^{k-1} = \det_k \left( \det_{n-k+1}(M^{S \backslash \{i\}}_{S \backslash \{j\}}) \right)_{i,j \in S}$

whenever ${n > k \geq 1}$, ${S}$ is a ${k}$-element subset of ${\{1,\dots,n\}}$, and ${M^S_{S'}}$ denotes the matrix formed from ${M}$ by removing the rows associated to ${S}$ and the columns associated to ${S'}$. (The Dodgson condensation identity is basically the ${k=2}$ case of this identity.)

A closely related proof of (2) proceeds by elementary row and column operations. Observe that if one adds some multiple of one of the first ${n-2}$ rows of ${M}$ to one of the last two rows of ${M}$, then the left and right sides of (2) do not change. If the minor ${A}$ is invertible, this allows one to reduce to the case where the components ${C_1,C_2}$ of the matrix vanish. Similarly, using elementary column operations instead of row operations we may assume that ${B_1,B_2}$ vanish. All matrices involved are now block-diagonal and the identity follows from a routine computation.

The latter approach can also prove the cute identity

$\displaystyle \det_2 \begin{pmatrix} \det_n( X_1, Y_1, A ) & \det_n( X_1, Y_2, A ) \\ \det_n(X_2, Y_1, A) & \det_n(X_2,Y_2, A) \end{pmatrix} = \det_n( X_1,X_2,A) \det_n(Y_1,Y_2,A)$

for any ${n \geq 2}$, any ${n \times 1}$ column vectors ${X_1,X_2,Y_1,Y_2}$, and any ${n \times n-2}$ matrix ${A}$, which can for instance be found in page 7 of this text of Karlin. Observe that both sides of this identity are unchanged if one adds some multiple of any column of ${A}$ to one of ${X_1,X_2,Y_1,Y_2}$; for generic ${A}$, this allows one to reduce ${X_1,X_2,Y_1,Y_2}$ to have only the first two entries allowed to be non-zero, at which point the determinants split into ${2 \times 2}$ and ${n -2 \times n-2}$ determinants and we can reduce to the ${n=2}$ case (eliminating the role of ${A}$). One can now either proceed by a direct computation, or by observing that the left-hand side is quartilinear in ${X_1,X_2,Y_1,Y_2}$ and antisymmetric in ${X_1,X_2}$ and ${Y_1,Y_2}$ which forces it to be a scalar multiple of ${\det_2(X_1,X_2) \det_2(Y_1,Y_2)}$, at which point one can test the identity at a single point (e.g. ${X_1=Y_1 = e_1}$ and ${X_2=Y_2=e_2}$ for the standard basis ${e_1,e_2}$) to conclude the argument. (One can also derive this identity from the Sylvester determinant identity but I think the calculations are a little messier if one goes by that route. Conversely, one can recover the Dodgson condensation identity from Karlin’s identity by setting ${X_1=e_1}$, ${X_2=e_2}$ (for instance) and then permuting some rows and columns.)

In one of the earliest posts on this blog, I talked about the ability to “arbitrage” a disparity of symmetry in an inequality, and in particular to “amplify” such an inequality into a stronger one. (The principle can apply to other mathematical statements than inequalities, with the “hypothesis” and “conclusion” of that statement generally playing the role of the “right-hand side” and “left-hand side” of an inequality, but for sake of discussion I will restrict attention here to inequalities.) One can formalise this principle as follows. Many inequalities in analysis can be expressed in the form

$\displaystyle A(f) \leq B(f) \ \ \ \ \ (1)$

for all ${f}$ in some space ${X}$ (in many cases ${X}$ will be a function space, and ${f}$ a function in that space), where ${A(f)}$ and ${B(f)}$ are some functionals of ${f}$ (that is to say, real-valued functions of ${f}$). For instance, ${B(f)}$ might be some function space norm of ${f}$ (e.g. an ${L^p}$ norm), and ${A(f)}$ might be some function space norm of some transform of ${f}$. In addition, we assume we have some group ${G}$ of symmetries ${T: X \rightarrow X}$ acting on the underlying space. For instance, if ${X}$ is a space of functions on some spatial domain, the group might consist of translations (e.g. ${Tf(x) = f(x-h)}$ for some shift ${h}$), or perhaps dilations with some normalisation (e.g. ${Tf(x) = \frac{1}{\lambda^\alpha} f(\frac{x}{\lambda})}$ for some dilation factor ${\lambda > 0}$ and some normalisation exponent ${\alpha \in {\bf R}}$, which can be thought of as the dimensionality of length one is assigning to ${f}$). If we have

$\displaystyle A(Tf) = A(f)$

for all symmetries ${T \in G}$ and all ${f \in X}$, we say that ${A}$ is invariant with respect to the symmetries in ${G}$; otherwise, it is not.

Suppose we know that the inequality (1) holds for all ${f \in X}$, but that there is an imbalance of symmetry: either ${A}$ is ${G}$-invariant and ${B}$ is not, or vice versa. Suppose first that ${A}$ is ${G}$-invariant and ${B}$ is not. Substituting ${f}$ by ${Tf}$ in (1) and taking infima, we can then amplify (1) to the stronger inequality

$\displaystyle A(f) \leq \inf_{T \in G} B(Tf).$

In particular, it is often the case that there is a way to send ${T}$ off to infinity in such a way that the functional ${B(Tf)}$ has a limit ${B_\infty(f)}$, in which case we obtain the amplification

$\displaystyle A(f) \leq B_\infty(f) \ \ \ \ \ (2)$

of (1). Note that these amplified inequalities will now be ${G}$-invariant on both sides (assuming that the way in which we take limits as ${T \rightarrow \infty}$ is itself ${G}$-invariant, which it often is in practice). Similarly, if ${B}$ is ${G}$-invariant but ${A}$ is not, we may instead amplify (1) to

$\displaystyle \sup_{T \in G} A(Tf) \leq B(f)$

and in particular (if ${A(Tf)}$ has a limit ${A_\infty(f)}$ as ${T \rightarrow \infty}$)

$\displaystyle A_\infty(f) \leq B(f). \ \ \ \ \ (3)$

If neither ${A(f)}$ nor ${B(f)}$ has a ${G}$-symmetry, one can still use the ${G}$-symmetry by replacing ${f}$ by ${Tf}$ and taking a limit to conclude that

$\displaystyle A_\infty(f) \leq B_\infty(f),$

though now this inequality is not obviously stronger than the original inequality (1) (for instance it could well be trivial). In some cases one can also average over ${G}$ instead of taking a limit as ${T \rightarrow \infty}$, thus averaging a non-invariant inequality into an invariant one.

As discussed in the previous post, this use of amplification gives rise to a general principle about inequalities: the most efficient inequalities are those in which the left-hand side and right-hand side enjoy the same symmetries. It is certainly possible to have true inequalities that have an imbalance of symmetry, but as shown above, such inequalities can always be amplified to more efficient and more symmetric inequalities. In the case when limits such as ${A_\infty}$ and ${B_\infty}$ exist, the limiting functionals ${A_\infty(f)}$ and ${B_\infty(f)}$ are often simpler in form, or more tractable analytically, than their non-limiting counterparts ${A(f)}$ and ${B(f)}$ (this is one of the main reasons why we take limits at infinity in the first place!), and so in many applications there is really no reason to use the weaker and more complicated inequality (1), when stronger, simpler, and more symmetric inequalities such as (2), (3) are available. Among other things, this explains why many of the most useful and natural inequalities one sees in analysis are dimensionally consistent.

One often tries to prove inequalities (1) by directly chaining together simpler inequalities. For instance, one might attempt to prove (1) by by first bounding ${A(f)}$ by some auxiliary quantity ${C(f)}$, and then bounding ${C(f)}$ by ${B(f)}$, thus obtaining (1) by chaining together two inequalities

$\displaystyle A(f) \leq C(f) \leq B(f). \ \ \ \ \ (4)$

A variant of the above principle then asserts that when proving inequalities by such direct methods, one should, whenever possible, try to maintain the symmetries that are present in both sides of the inequality. Why? Well, suppose that we ignored this principle and tried to prove (1) by establishing (4) for some ${C}$ that is not ${G}$-invariant. Assuming for sake of argument that (4) were actually true, we could amplify the first half ${A(f) \leq C(f)}$ of this inequality to conclude that

$\displaystyle A(f) \leq \inf_{T \in G} C(Tf)$

and also amplify the second half ${C(f) \leq B(f)}$ of the inequality to conclude that

$\displaystyle \sup_{T \in G} C(Tf) \leq B(f)$

and hence (4) amplifies to

$\displaystyle A(f) \leq \inf_{T \in G} C(Tf) \leq \sup_{T \in G} C(Tf) \leq B(f). \ \ \ \ \ (5)$

Let’s say for sake of argument that all the quantities involved here are positive numbers (which is often the case in analysis). Then we see in particular that

$\displaystyle \frac{\sup_{T \in G} C(Tf)}{\inf_{T \in G} C(Tf)} \leq \frac{B(f)}{A(f)}. \ \ \ \ \ (6)$

Informally, (6) asserts that in order for the strategy (4) used to prove (1) to work, the extent to which ${C}$ fails to be ${G}$-invariant cannot exceed the amount of “room” present in (1). In particular, when dealing with those “extremal” ${f}$ for which the left and right-hand sides of (1) are comparable to each other, one can only have a bounded amount of non-${G}$-invariance in the functional ${C}$. If ${C}$ fails so badly to be ${G}$-invariant that one does not expect the left-hand side of (6) to be at all bounded in such extremal situations, then the strategy of proving (1) using the intermediate quantity ${C}$ is doomed to failure – even if one has already produced some clever proof of one of the two inequalities ${A(f) \leq C(f)}$ or ${C(f) \leq B(f)}$ needed to make this strategy work. And even if it did work, one could amplify (4) to a simpler inequality

$\displaystyle A(f) \leq C_\infty(f) \leq B(f) \ \ \ \ \ (7)$

(assuming that the appropriate limit ${C_\infty(f) = \lim_{T \rightarrow \infty} C(Tf)}$ existed) which would likely also be easier to prove (one can take whatever proofs one had in mind of the inequalities in (4), conjugate them by ${T}$, and take a limit as ${T \rightarrow \infty}$ to extract a proof of (7)).

Here are some simple (but somewhat contrived) examples to illustrate these points. Suppose one wishes to prove the inequality

$\displaystyle xy \leq x^2 + y^2 \ \ \ \ \ (8)$

for all ${x,y>0}$. Both sides of this inequality are invariant with respect to interchanging ${x}$ with ${y}$, so the principle suggests that when proving this inequality directly, one should only use sub-inequalities that are also invariant with respect to this interchange. However, in this particular case there is enough “room” in the inequality that it is possible (though somewhat unnatural) to violate this principle. For instance, one could decide (for whatever reason) to start with the inequality

$\displaystyle 0 \leq (x - y/2)^2 = x^2 - xy + y^2/4$

to conclude that

$\displaystyle xy \leq x^2 + y^2/4$

and then use the obvious inequality ${x^2 + y^2/4 \leq x^2+y^2}$ to conclude the proof. Here, the intermediate quantity ${x^2 + y^2/4}$ is not invariant with respect to interchange of ${x}$ and ${y}$, but the failure is fairly mild (changing ${x}$ and ${y}$ only modifies the quantity ${x^2 + y^2/4}$ by a multiplicative factor of ${4}$ at most), and disappears completely in the most extremal case ${x=y}$, which helps explain why one could get away with using this quantity in the proof here. But it would be significantly harder (though still not impossible) to use non-symmetric intermediaries to prove the sharp version

$\displaystyle xy \leq \frac{x^2 + y^2}{2}$

of (8) (that is to say, the arithmetic mean-geometric mean inequality). Try it!

Similarly, consider the task of proving the triangle inequality

$\displaystyle |z+w| \leq |z| + |w| \ \ \ \ \ (9)$

for complex numbers ${z, w}$. One could try to leverage the triangle inequality ${|x+y| \leq |x| + |y|}$ for real numbers by using the crude estimate

$\displaystyle |z+w| \leq |\hbox{Re}(z+w)| + |\hbox{Im}(z+w)|$

and then use the real triangle inequality to obtain

$\displaystyle |\hbox{Re}(z+w)| \leq |\hbox{Re}(z)| + |\hbox{Re}(w)|$

and

$\displaystyle |\hbox{Im}(z+w)| \leq |\hbox{Im}(z)| + |\hbox{Im}(w)|$

and then finally use the inequalities

$\displaystyle |\hbox{Re}(z)|, |\hbox{Im}(z)| \leq |z| \ \ \ \ \ (10)$

and

$\displaystyle |\hbox{Re}(w)|, |\hbox{Im}(w)| \leq |w| \ \ \ \ \ (11)$

but when one puts this all together at the end of the day, one loses a factor of two:

$\displaystyle |z+w| \leq 2(|z| + |w|).$

One can “blame” this loss on the fact that while the original inequality (9) was invariant with respect to phase rotation ${(z,w) \mapsto (e^{i\theta} z, e^{i\theta} w)}$, the intermediate expressions we tried to use when proving it were not, leading to inefficient estimates. One can try to be smarter than this by using Pythagoras’ theorem ${|z|^2 = |\hbox{Re}(z)|^2 + |\hbox{Im}(z)|^2}$; this reduces the loss from ${2}$ to ${\sqrt{2}}$ but does not eliminate it completely, which is to be expected as one is still using non-invariant estimates in the proof. But one can remove the loss completely by using amplification; see the previous blog post for details (we also give a reformulation of this amplification below).

Here is a slight variant of the above example. Suppose that you had just learned in class to prove the triangle inequality

$\displaystyle (\sum_{n=1}^\infty |a_n+b_n|^2)^{1/2} \leq (\sum_{n=1}^\infty |a_n|^2)^{1/2} + (\sum_{n=1}^\infty |b_n|^2)^{1/2} \ \ \ \ \ (12)$

for (say) real square-summable sequences ${(a_n)_{n=1}^\infty}$, ${(b_n)_{n=1}^\infty}$, and was tasked to conclude the corresponding inequality

$\displaystyle (\sum_{n \in {\bf Z}} |a_n+b_n|^2)^{1/2} \leq (\sum_{n \in {\bf Z}} |a_n|^2)^{1/2} + (\sum_{n \in {\bf Z}} |b_n|^2)^{1/2} \ \ \ \ \ (13)$

for doubly infinite square-summable sequences ${(a_n)_{n \in {\bf Z}}, (b_n)_{n \in {\bf Z}}}$. The quickest way to do this is of course to exploit a bijection between the natural numbers ${1,2,\dots}$ and the integers, but let us say for sake of argument that one was unaware of such a bijection. One could then proceed instead by splitting the integers into the positive integers and the non-positive integers, and use (12) on each component separately; this is very similar to the strategy of proving (9) by splitting a complex number into real and imaginary parts, and will similarly lose a factor of ${2}$ or ${\sqrt{2}}$. In this case, one can “blame” this loss on the abandonment of translation invariance: both sides of the inequality (13) are invariant with respect to shifting the sequences ${(a_n)_{n \in {\bf Z}}}$, ${(b_n)_{n \in {\bf Z}}}$ by some shift ${h}$ to arrive at ${(a_{n-h})_{n \in {\bf Z}}, (b_{n-h})_{n \in {\bf Z}}}$, but the intermediate quantities caused by splitting the integers into two subsets are not invariant. Another way of thinking about this is that the splitting of the integers gives a privileged role to the origin ${n=0}$, whereas the inequality (13) treats all values of ${n}$ equally thanks to the translation invariance, and so using such a splitting is unnatural and not likely to lead to optimal estimates. On the other hand, one can deduce (13) from (12) by sending this symmetry to infinity; indeed, after applying a shift to (12) we see that

$\displaystyle (\sum_{n=-N}^\infty |a_n+b_n|^2)^{1/2} \leq (\sum_{n=-N}^\infty |a_n|^2)^{1/2} + (\sum_{n=-N}^\infty |b_n|^2)^{1/2}$

for any ${N}$, and on sending ${N \rightarrow \infty}$ we obtain (13) (one could invoke the monotone convergence theorem here to justify the limit, though in this case it is simple enough that one can just use first principles).

Note that the principle of preserving symmetry only applies to direct approaches to proving inequalities such as (1). There is a complementary approach, discussed for instance in this previous post, which is to spend the symmetry to place the variable ${f}$ “without loss of generality” in a “normal form”, “convenient coordinate system”, or a “good gauge”. Abstractly: suppose that there is some subset ${Y}$ of ${X}$ with the property that every ${f \in X}$ can be expressed in the form ${f = Tg}$ for some ${T \in G}$ and ${g \in Y}$ (that is to say, ${X = GY}$). Then, if one wishes to prove an inequality (1) for all ${f \in X}$, and one knows that both sides ${A(f), B(f)}$ of this inequality are ${G}$-invariant, then it suffices to check (1) just for those ${f}$ in ${Y}$, as this together with the ${G}$-invariance will imply the same inequality (1) for all ${f}$ in ${GY=X}$. By restricting to those ${f}$ in ${Y}$, one has given up (or spent) the ${G}$-invariance, as the set ${Y}$ will in typical not be preserved by the group action ${G}$. But by the same token, by eliminating the invariance, one also eliminates the prohibition on using non-invariant proof techniques, and one is now free to use a wider range of inequalities in order to try to establish (1). Of course, such inequalities should make crucial use of the restriction ${f \in Y}$, for if they did not, then the arguments would work in the more general setting ${f \in X}$, and then the previous principle would again kick in and warn us that the use of non-invariant inequalities would be inefficient. Thus one should “spend” the symmetry wisely to “buy” a restriction ${f \in Y}$ that will be of maximal utility in calculations (for instance by setting as many annoying factors and terms in one’s analysis to be ${0}$ or ${1}$ as possible).

As a simple example of this, let us revisit the complex triangle inequality (9). As already noted, both sides of this inequality are invariant with respect to the phase rotation symmetry ${(z,w) \mapsto (e^{i\theta} z, e^{i\theta} w)}$. This seems to limit one to using phase-rotation-invariant techniques to establish the inequality, in particular ruling out the use of real and imaginary parts as discussed previously. However, we can instead spend the phase rotation symmetry to restrict to a special class of ${z}$ and ${w}$. It turns out that the most efficient way to spend the symmetry is to achieve the normalisation of ${z+w}$ being a nonnegative real; this is of course possible since any complex number ${z+w}$ can be turned into a nonnegative real by multiplying by an appropriate phase ${e^{i\theta}}$. Once ${z+w}$ is a nonnegative real, the imaginary part disappears and we have

$\displaystyle |z+w| = \hbox{Re}(z+w) = \hbox{Re}(z) + \hbox{Re}(w),$

and the triangle inequality (9) is now an immediate consequence of (10), (11). (But note that if one had unwisely spent the symmetry to normalise, say, ${z}$ to be a non-negative real, then one is no closer to establishing (9) than before one had spent the symmetry.)

Apoorva Khare and I have just uploaded to the arXiv our paper “On the sign patterns of entrywise positivity preservers in fixed dimension“. This paper explores the relationship between positive definiteness of Hermitian matrices, and entrywise operations on these matrices. The starting point for this theory is the Schur product theorem, which asserts that if ${A = (a_{ij})_{1 \leq i,j \leq N}}$ and ${B = (b_{ij})_{1 \leq i,j \leq N}}$ are two ${N \times N}$ Hermitian matrices that are positive semi-definite, then their Hadamard product

$\displaystyle A \circ B := (a_{ij} b_{ij})_{1 \leq i,j \leq N}$

is also positive semi-definite. (One should caution that the Hadamard product is not the same as the usual matrix product.) To prove this theorem, first observe that the claim is easy when ${A = {\bf u} {\bf u}^*}$ and ${B = {\bf v} {\bf v}^*}$ are rank one positive semi-definite matrices, since in this case ${A \circ B = ({\bf u} \circ {\bf v}) ({\bf u} \circ {\bf v})^*}$ is also a rank one positive semi-definite matrix. The general case then follows by noting from the spectral theorem that a general positive semi-definite matrix can be expressed as a non-negative linear combination of rank one positive semi-definite matrices, and using the bilinearity of the Hadamard product and the fact that the set of positive semi-definite matrices form a convex cone. A modification of this argument also lets one replace “positive semi-definite” by “positive definite” in the statement of the Schur product theorem.

One corollary of the Schur product theorem is that any polynomial ${P(z) = c_0 + c_1 z + \dots + c_d z^d}$ with non-negative coefficients ${c_n \geq 0}$ is entrywise positivity preserving on the space ${{\mathbb P}_N({\bf C})}$ of ${N \times N}$ positive semi-definite Hermitian matrices, in the sense that for any matrix ${A = (a_{ij})_{1 \leq i,j \leq N}}$ in ${{\mathbb P}_N({\bf C})}$, the entrywise application

$\displaystyle P[A] := (P(a_{ij}))_{1 \leq i,j \leq N}$

of ${P}$ to ${A}$ is also positive semi-definite. (As before, one should caution that ${P[A]}$ is not the application ${P(A)}$ of ${P}$ to ${A}$ by the usual functional calculus.) Indeed, one can expand

$\displaystyle P[A] = c_0 A^{\circ 0} + c_1 A^{\circ 1} + \dots + c_d A^{\circ d},$

where ${A^{\circ i}}$ is the Hadamard product of ${i}$ copies of ${A}$, and the claim now follows from the Schur product theorem and the fact that ${{\mathbb P}_N({\bf C})}$ is a convex cone.

A slight variant of this argument, already observed by Pólya and Szegö in 1925, shows that if ${I}$ is any subset of ${{\bf C}}$ and

$\displaystyle f(z) = \sum_{n=0}^\infty c_n z^n \ \ \ \ \ (1)$

is a power series with non-negative coefficients ${c_n \geq 0}$ that is absolutely and uniformly convergent on ${I}$, then ${f}$ will be entrywise positivity preserving on the set ${{\mathbb P}_N(I)}$ of positive definite matrices with entries in ${I}$. (In the case that ${I}$ is of the form ${I = [0,\rho]}$, such functions are precisely the absolutely monotonic functions on ${I}$.)

In the work of Schoenberg and of Rudin, we have a converse: if ${f: (-1,1) \rightarrow {\bf C}}$ is a function that is entrywise positivity preserving on ${{\mathbb P}_N((-1,1))}$ for all ${N}$, then it must be of the form (1) with ${c_n \geq 0}$. Variants of this result, with ${(-1,1)}$ replaced by other domains, appear in the work of Horn, Vasudeva, and Guillot-Khare-Rajaratnam.

This gives a satisfactory classification of functions ${f}$ that are entrywise positivity preservers in all dimensions ${N}$ simultaneously. However, the question remains as to what happens if one fixes the dimension ${N}$, in which case one may have a larger class of entrywise positivity preservers. For instance, in the trivial case ${N=1}$, a function would be entrywise positivity preserving on ${{\mathbb P}_1((0,\rho))}$ if and only if ${f}$ is non-negative on ${I}$. For higher ${N}$, there is a necessary condition of Horn (refined slightly by Guillot-Khare-Rajaratnam) which asserts (at least in the case of smooth ${f}$) that all derivatives of ${f}$ at zero up to ${(N-1)^{th}}$ order must be non-negative in order for ${f}$ to be entrywise positivity preserving on ${{\mathbb P}_N((0,\rho))}$ for some ${0 < \rho < \infty}$. In particular, if ${f}$ is of the form (1), then ${c_0,\dots,c_{N-1}}$ must be non-negative. In fact, a stronger assertion can be made, namely that the first ${N}$ non-zero coefficients in (1) (if they exist) must be positive, or equivalently any negative term in (1) must be preceded (though not necessarily immediately) by at least ${N}$ positive terms. If ${f}$ is of the form (1) is entrywise positivity preserving on the larger set ${{\mathbb P}_N((0,+\infty))}$, one can furthermore show that any negative term in (1) must also be followed (though not necessarily immediately) by at least ${N}$ positive terms.

The main result of this paper is that these sign conditions are the only constraints for entrywise positivity preserving power series. More precisely:

Theorem 1 For each ${n}$, let ${\epsilon_n \in \{-1,0,+1\}}$ be a sign.

• Suppose that any negative sign ${\epsilon_M = -1}$ is preceded by at least ${N}$ positive signs (thus there exists ${0 \leq n_0 < \dots < n_{N-1}< M}$ with ${\epsilon_{n_0} = \dots = \epsilon_{n_{N-1}} = +1}$). Then, for any ${0 < \rho < \infty}$, there exists a convergent power series (1) on ${(0,\rho)}$, with each ${c_n}$ having the sign of ${\epsilon_n}$, which is entrywise positivity preserving on ${{\bf P}_N((0,\rho))}$.
• Suppose in addition that any negative sign ${\epsilon_M = -1}$ is followed by at least ${N}$ positive signs (thus there exists ${M < n_N < \dots < n_{2N-1}}$ with ${\epsilon_{n_N} = \dots = \epsilon_{n_{2N-1}} = +1}$). Then there exists a convergent power series (1) on ${(0,+\infty)}$, with each ${c_n}$ having the sign of ${\epsilon_n}$, which is entrywise positivity preserving on ${{\bf P}_N((0,+\infty))}$.

One can ask the same question with ${(0,\rho)}$ or ${(0,+\infty)}$ replaced by other domains such as ${(-\rho,\rho)}$, or the complex disk ${D(0,\rho)}$, but it turns out that there are far fewer entrywise positivity preserving functions in those cases basically because of the non-trivial zeroes of Schur polynomials in these ranges; see the paper for further discussion. We also have some quantitative bounds on how negative some of the coefficients can be compared to the positive coefficients, but they are a bit technical to state here.

The heart of the proofs of these results is an analysis of the determinants ${\mathrm{det} P[ {\bf u} {\bf u}^* ]}$ of polynomials ${P}$ applied entrywise to rank one matrices ${uu^*}$; the positivity of these determinants can be used (together with a continuity argument) to establish the positive definiteness of ${P[uu^*]}$ for various ranges of ${P}$ and ${u}$. Using the Cauchy-Binet formula, one can rewrite such determinants as linear combinations of squares of magnitudes of generalised Vandermonde determinants

$\displaystyle \mathrm{det}( u_i^{n_j} )_{1 \leq i,j \leq N},$

where ${{\bf u} = (u_1,\dots,u_N)}$ and the signs of the coefficients in the linear combination are determined by the signs of the coefficients of ${P}$. The task is then to find upper and lower bounds for the magnitudes of such generalised Vandermonde determinants. These determinants oscillate in sign, which makes the problem look difficult; however, an algebraic miracle intervenes, namely the factorisation

$\displaystyle \mathrm{det}( u_i^{n_j} )_{1 \leq i,j \leq N} = \pm V({\bf u}) s_\lambda({\bf u})$

of the generalised Vandermonde determinant into the ordinary Vandermonde determinant

$\displaystyle V({\bf u}) = \prod_{1 \leq i < j \leq N} (u_j - u_i)$

and a Schur polynomial ${s_\lambda}$ applied to ${{\bf u}}$, where the weight ${\lambda}$ of the Schur polynomial is determined by ${n_1,\dots,n_N}$ in a simple fashion. The problem then boils down to obtaining upper and lower bounds for these Schur polynomials. Because we are restricting attention to matrices taking values in ${(0,\rho)}$ or ${(0,+\infty)}$, the entries of ${{\bf u}}$ can be taken to be non-negative. One can then take advantage of the total positivity of the Schur polynomials to compare these polynomials with a monomial, at which point one can obtain good criteria for ${P[A]}$ to be positive definite when ${A}$ is a rank one positive definite matrix ${A = {\bf u} {\bf u}^*}$.

If we allow the exponents ${n_1,\dots,n_N}$ to be real numbers rather than integers (thus replacing polynomials or power series by Pusieux series or Hahn series), then we lose the above algebraic miracle, but we can replace it with a geometric miracle, namely the Harish-Chandra-Itzykson-Zuber identity, which I discussed in this previous blog post. This factors the above generalised Vandermonde determinant as the product of the ordinary Vandermonde determinant and an integral of a positive quantity over the orthogonal group, which one can again compare with a monomial after some fairly elementary estimates.

It remains to understand what happens for more general positive semi-definite matrices ${A}$. Here we use a trick of FitzGerald and Horn to amplify the rank one case to the general case, by expressing a general positive semi-definite matrix ${A}$ as a linear combination of a rank one matrix ${{\bf u} {\bf u}^*}$ and another positive semi-definite matrix ${B}$ that vanishes on the last row and column (and is thus effectively a positive definite ${N-1 \times N-1}$ matrix). Using the fundamental theorem of calculus to continuously deform the rank one matrix ${{\bf u} {\bf u}^*}$ to ${A}$ in the direction ${B}$, one can then obtain positivity results for ${P[A]}$ from positivity results for ${P[{\bf u} {\bf u}^*]}$ combined with an induction hypothesis on ${N}$.

Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures“, submitted to Duke Mathematical Journal. This paper builds upon my previous paper in which I introduced an “entropy decrement method” to prove the two-point (logarithmically averaged) cases of the Chowla and Elliott conjectures. A bit more specifically, I showed that

$\displaystyle \lim_{m \rightarrow \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_0(n+h_0) g_1(n+h_1)}{n} = 0$

whenever ${1 \leq \omega_m \leq x_m}$ were sequences going to infinity, ${h_0,h_1}$ were distinct integers, and ${g_0,g_1: {\bf N} \rightarrow {\bf C}}$ were ${1}$-bounded multiplicative functions which were non-pretentious in the sense that

$\displaystyle \liminf_{X \rightarrow \infty} \inf_{|t_j| \leq X} \sum_{p \leq X} \frac{1-\mathrm{Re}( g_j(p) \overline{\chi_j}(p) p^{it_j})}{p} = \infty \ \ \ \ \ (1)$

for all Dirichlet characters ${\chi_j}$ and for ${j=0,1}$. Thus, for instance, one had the logarithmically averaged two-point Chowla conjecture

$\displaystyle \sum_{n \leq x} \frac{\lambda(n) \lambda(n+h)}{n} = o(\log x)$

for fixed any non-zero ${h}$, where ${\lambda}$ was the Liouville function.

One would certainly like to extend these results to higher order correlations than the two-point correlations. This looks to be difficult (though perhaps not completely impossible if one allows for logarithmic averaging): in a previous paper I showed that achieving this in the context of the Liouville function would be equivalent to resolving the logarithmically averaged Sarnak conjecture, as well as establishing logarithmically averaged local Gowers uniformity of the Liouville function. However, in this paper we are able to avoid having to resolve these difficult conjectures to obtain partial results towards the (logarithmically averaged) Chowla and Elliott conjecture. For the Chowla conjecture, we can obtain all odd order correlations, in that

$\displaystyle \sum_{n \leq x} \frac{\lambda(n+h_1) \dots \lambda(n+h_k)}{n} = o(\log x) \ \ \ \ \ (2)$

for all odd ${k}$ and all integers ${h_1,\dots,h_k}$ (which, in the odd order case, are no longer required to be distinct). (Superficially, this looks like we have resolved “half of the cases” of the logarithmically averaged Chowla conjecture; but it seems the odd order correlations are significantly easier than the even order ones. For instance, because of the Katai-Bourgain-Sarnak-Ziegler criterion, one can basically deduce the odd order cases of (2) from the even order cases (after allowing for some dilations in the argument ${n}$).

For the more general Elliott conjecture, we can show that

$\displaystyle \lim_{m \rightarrow \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_1(n+h_1) \dots g_k(n+h_k)}{n} = 0$

for any ${k}$, any integers ${h_1,\dots,h_k}$ and any bounded multiplicative functions ${g_1,\dots,g_k}$, unless the product ${g_1 \dots g_k}$ weakly pretends to be a Dirichlet character ${\chi}$ in the sense that

$\displaystyle \sum_{p \leq X} \frac{1 - \hbox{Re}( g_1 \dots g_k(p) \overline{\chi}(p)}{p} = o(\log\log X).$

This can be seen to imply (2) as a special case. Even when ${g_1,\dots,g_k}$ does pretend to be a Dirichlet character ${\chi}$, we can still say something: if the limits

$\displaystyle f(a) := \lim_{m \rightarrow \infty} \frac{1}{\log \omega_m} \sum_{x_m/\omega_m \leq n \leq x_m} \frac{g_1(n+ah_1) \dots g_k(n+ah_k)}{n}$

exist for each ${a \in {\bf Z}}$ (which can be guaranteed if we pass to a suitable subsequence), then ${f}$ is the uniform limit of periodic functions ${f_i}$, each of which is ${\chi}$isotypic in the sense that ${f_i(ab) = f_i(a) \chi(b)}$ whenever ${a,b}$ are integers with ${b}$ coprime to the periods of ${\chi}$ and ${f_i}$. This does not pin down the value of any single correlation ${f(a)}$, but does put significant constraints on how these correlations may vary with ${a}$.

Among other things, this allows us to show that all ${16}$ possible length four sign patterns ${(\lambda(n+1),\dots,\lambda(n+4)) \in \{-1,+1\}^4}$ of the Liouville function occur with positive density, and all ${65}$ possible length four sign patterns ${(\mu(n+1),\dots,\mu(n+4)) \in \{-1,0,+1\}^4 \backslash \{-1,+1\}^4}$ occur with the conjectured logarithmic density. (In a previous paper with Matomaki and Radziwill, we obtained comparable results for length three patterns of Liouville and length two patterns of Möbius.)

To describe the argument, let us focus for simplicity on the case of the Liouville correlations

$\displaystyle f(a) := \lim_{X \rightarrow \infty} \frac{1}{\log X} \sum_{n \leq X} \frac{\lambda(n) \lambda(n+a) \dots \lambda(n+(k-1)a)}{n}, \ \ \ \ \ (3)$

assuming for sake of discussion that all limits exist. (In the paper, we instead use the device of generalised limits, as discussed in this previous post.) The idea is to combine together two rather different ways to control this function ${f}$. The first proceeds by the entropy decrement method mentioned earlier, which roughly speaking works as follows. Firstly, we pick a prime ${p}$ and observe that ${\lambda(pn)=-\lambda(n)}$ for any ${n}$, which allows us to rewrite (3) as

$\displaystyle (-1)^k f(a) = \lim_{X \rightarrow \infty} \frac{1}{\log X}$

$\displaystyle \sum_{n \leq X} \frac{\lambda(pn) \lambda(pn+ap) \dots \lambda(pn+(k-1)ap)}{n}.$

Making the change of variables ${n' = pn}$, we obtain

$\displaystyle (-1)^k f(a) = \lim_{X \rightarrow \infty} \frac{1}{\log X}$

$\displaystyle \sum_{n' \leq pX} \frac{\lambda(n') \lambda(n'+ap) \dots \lambda(n'+(k-1)ap)}{n'} p 1_{p|n'}.$

The difference between ${n' \leq pX}$ and ${n' \leq X}$ is negligible in the limit (here is where we crucially rely on the log-averaging), hence

$\displaystyle (-1)^k f(a) = \lim_{X \rightarrow \infty} \frac{1}{\log X} \sum_{n \leq X} \frac{\lambda(n) \lambda(n+ap) \dots \lambda(n+(k-1)ap)}{n} p 1_{p|n}$

and thus by (3) we have

$\displaystyle (-1)^k f(a) = f(ap) + \lim_{X \rightarrow \infty} \frac{1}{\log X}$

$\displaystyle \sum_{n \leq X} \frac{\lambda(n) \lambda(n+ap) \dots \lambda(n+(k-1)ap)}{n} (p 1_{p|n}-1).$

The entropy decrement argument can be used to show that the latter limit is small for most ${p}$ (roughly speaking, this is because the factors ${p 1_{p|n}-1}$ behave like independent random variables as ${p}$ varies, so that concentration of measure results such as Hoeffding’s inequality can apply, after using entropy inequalities to decouple somewhat these random variables from the ${\lambda}$ factors). We thus obtain the approximate isotopy property

$\displaystyle (-1)^k f(a) \approx f(ap) \ \ \ \ \ (4)$

for most ${a}$ and ${p}$.

On the other hand, by the Furstenberg correspondence principle (as discussed in these previous posts), it is possible to express ${f(a)}$ as a multiple correlation

$\displaystyle f(a) = \int_X g(x) g(T^a x) \dots g(T^{(k-1)a} x)\ d\mu(x)$

for some probability space ${(X,\mu)}$ equipped with a measure-preserving invertible map ${T: X \rightarrow X}$. Using results of Bergelson-Host-Kra, Leibman, and Le, this allows us to obtain a decomposition of the form

$\displaystyle f(a) = f_1(a) + f_2(a) \ \ \ \ \ (5)$

where ${f_1}$ is a nilsequence, and ${f_2}$ goes to zero in density (even along the primes, or constant multiples of the primes). The original work of Bergelson-Host-Kra required ergodicity on ${X}$, which is very definitely a hypothesis that is not available here; however, the later work of Leibman removed this hypothesis, and the work of Le refined the control on ${f_1}$ so that one still has good control when restricting to primes, or constant multiples of primes.

Ignoring the small error ${f_2(a)}$, we can now combine (5) to conclude that

$\displaystyle f(a) \approx (-1)^k f_1(ap).$

Using the equidistribution theory of nilsequences (as developed in this previous paper of Ben Green and myself), one can break up ${f_1}$ further into a periodic piece ${f_0}$ and an “irrational” or “minor arc” piece ${f_3}$. The contribution of the minor arc piece ${f_3}$ can be shown to mostly cancel itself out after dilating by primes ${p}$ and averaging, thanks to Vinogradov-type bilinear sum estimates (transferred to the primes). So we end up with

$\displaystyle f(a) \approx (-1)^k f_0(ap),$

which already shows (heuristically, at least) the claim that ${f}$ can be approximated by periodic functions ${f_0}$ which are isotopic in the sense that

$\displaystyle f_0(a) \approx (-1)^k f_0(ap).$

But if ${k}$ is odd, one can use Dirichlet’s theorem on primes in arithmetic progressions to restrict to primes ${p}$ that are ${1}$ modulo the period of ${f_0}$, and conclude now that ${f_0}$ vanishes identically, which (heuristically, at least) gives (2).

The same sort of argument works to give the more general bounds on correlations of bounded multiplicative functions. But for the specific task of proving (2), we initially used a slightly different argument that avoids using the ergodic theory machinery of Bergelson-Host-Kra, Leibman, and Le, but replaces it instead with the Gowers uniformity norm theory used to count linear equations in primes. Basically, by averaging (4) in ${p}$ using the “${W}$-trick”, as well as known facts about the Gowers uniformity of the von Mangoldt function, one can obtain an approximation of the form

$\displaystyle (-1)^k f(a) \approx {\bf E}_{b: (b,W)=1} f(ab)$

where ${b}$ ranges over a large range of integers coprime to some primorial ${W = \prod_{p \leq w} p}$. On the other hand, by iterating (4) we have

$\displaystyle f(a) \approx f(apq)$

for most semiprimes ${pq}$, and by again averaging over semiprimes one can obtain an approximation of the form

$\displaystyle f(a) \approx {\bf E}_{b: (b,W)=1} f(ab).$

For ${k}$ odd, one can combine the two approximations to conclude that ${f(a)=0}$. (This argument is not given in the current paper, but we plan to detail it in a subsequent one.)

The complete homogeneous symmetric polynomial ${h_d(x_1,\dots,x_n)}$ of ${n}$ variables ${x_1,\dots,x_n}$ and degree ${d}$ can be defined as

$\displaystyle h_d(x_1,\dots,x_n) := \sum_{1 \leq i_1 \leq \dots \leq i_d \leq n} x_{i_1} \dots x_{i_d},$

thus for instance

$\displaystyle h_0(x_1,\dots,x_n) = 0,$

$\displaystyle h_1(x_1,\dots,x_n) = x_1 + \dots + x_n,$

and

$\displaystyle h_2(x_1,\dots,x_n) = x_1^2 + \dots + x_n^2 + \sum_{1 \leq i < j \leq n} x_i x_j.$

One can also define all the complete homogeneous symmetric polynomials of ${n}$ variables simultaneously by means of the generating function

$\displaystyle \sum_{d=0}^\infty h_d(x_1,\dots,x_n) t^d = \frac{1}{(1-t x_1) \dots (1-tx_n)}. \ \ \ \ \ (1)$

We will think of the variables ${x_1,\dots,x_n}$ as taking values in the real numbers. When one does so, one might observe that the degree two polynomial ${h_2}$ is a positive definite quadratic form, since it has the sum of squares representation

$\displaystyle h_2(x_1,\dots,x_n) = \frac{1}{2} \sum_{i=1}^n x_i^2 + \frac{1}{2} (x_1+\dots+x_n)^2.$

In particular, ${h_2(x_1,\dots,x_n) > 0}$ unless ${x_1=\dots=x_n=0}$. This can be compared against the superficially similar quadratic form

$\displaystyle x_1^2 + \dots + x_n^2 + \sum_{1 \leq i < j \leq n} \epsilon_{ij} x_i x_j$

where ${\epsilon_{ij} = \pm 1}$ are independent randomly chosen signs. The Wigner semicircle law says that for large ${n}$, the eigenvalues of this form will be mostly distributed in the interval ${[-\sqrt{n}, \sqrt{n}]}$ using the semicircle distribution, so in particular the form is quite far from being positive definite despite the presence of the first ${n}$ positive terms. Thus the positive definiteness is coming from the finer algebraic structure of ${h_2}$, and not just from the magnitudes of its coefficients.

One could ask whether the same positivity holds for other degrees ${d}$ than two. For odd degrees, the answer is clearly no, since ${h_d(-x_1,\dots,-x_n) = -h_d(x_1,\dots,x_n)}$ in that case. But one could hope for instance that

$\displaystyle h_4(x_1,\dots,x_n) = \sum_{1 \leq i \leq j \leq k \leq l \leq n} x_i x_j x_k x_l$

also has a sum of squares representation that demonstrates positive definiteness. This turns out to be true, but is remarkably tedious to establish directly. Nevertheless, we have a nice result of Hunter that gives positive definiteness for all even degrees ${d}$. In fact, a modification of his argument gives a little bit more:

Theorem 1 Let ${n \geq 1}$, let ${d \geq 0}$ be even, and let ${x_1,\dots,x_n}$ be reals.

• (i) (Positive definiteness) One has ${h_d(x_1,\dots,x_n) \geq 0}$, with strict inequality unless ${x_1=\dots=x_n=0}$.
• (ii) (Schur convexity) One has ${h_d(x_1,\dots,x_n) \geq h_d(y_1,\dots,y_n)}$ whenever ${(x_1,\dots,x_n)}$ majorises ${(y_1,\dots,y_n)}$, with equality if and only if ${(y_1,\dots,y_n)}$ is a permutation of ${(x_1,\dots,x_n)}$.
• (iii) (Schur-Ostrowski criterion for Schur convexity) For any ${1 \leq i < j \leq n}$, one has ${(x_i - x_j) (\frac{\partial}{\partial x_i} - \frac{\partial}{\partial x_j}) h_d(x_1,\dots,x_n) \geq 0}$, with strict inequality unless ${x_i=x_j}$.

Proof: We induct on ${d}$ (allowing ${n}$ to be arbitrary). The claim is trivially true for ${d=0}$, and easily verified for ${d=2}$, so suppose that ${d \geq 4}$ and the claims (i), (ii), (iii) have already been proven for ${d-2}$ (and for arbitrary ${n}$).

If we apply the differential operator ${(x_i - x_j) (\frac{\partial}{\partial x_i} - \frac{\partial}{\partial x_j})}$ to ${\frac{1}{(1-t x_1) \dots (1-tx_n)}}$ using the product rule, one obtains after a brief calculation

$\displaystyle \frac{(x_i-x_j)^2 t^2}{(1-t x_1) \dots (1-tx_n) (1-t x_i) (1-t x_j)}.$

Using (1) and extracting the ${t^d}$ coefficient, we obtain the identity

$\displaystyle (x_i - x_j) (\frac{\partial}{\partial x_i} - \frac{\partial}{\partial x_j}) h_d(x_1,\dots,x_n)$

$\displaystyle = (x_i-x_j)^2 h_{d-2}( x_1,\dots,x_n,x_i,x_j). \ \ \ \ \ (2)$

The claim (iii) then follows from (i) and the induction hypothesis.

To obtain (ii), we use the more general statement (known as the Schur-Ostrowski criterion) that (ii) is implied from (iii) if we replace ${h_d}$ by an arbitrary symmetric, continuously differentiable function. To establish this criterion, we induct on ${n}$ (this argument can be made independently of the existing induction on ${d}$). If ${(y_1,\dots,y_n)}$ is majorised by ${(x_1,\dots,x_n)}$, it lies in the permutahedron of ${(x_1,\dots,x_n)}$. If ${(y_1,\dots,y_n)}$ lies on a face of this permutahedron, then after permuting both the ${x_i}$ and ${y_j}$ we may assume that ${(y_1,\dots,y_m)}$ is majorised by ${(x_1,\dots,x_m)}$, and ${(y_{m+1},\dots,y_n)}$ is majorised by ${(x_{m+1},\dots,x_n)}$ for some ${1 \leq m < n}$, and the claim then follows from two applications of the induction hypothesis. If instead ${(y_1,\dots,y_n)}$ lies in the interior of the permutahedron, one can follow it to the boundary by using one of the vector fields ${(x_i - x_j) (\frac{\partial}{\partial x_i} - \frac{\partial}{\partial x_j})}$, and the claim follows from the boundary case.

Finally, to obtain (i), we observe that ${(x_1,\dots,x_n)}$ majorises ${(x,\dots,x)}$, where ${x}$ is the arithmetic mean of ${x_1,\dots,x_n}$. But ${h_d(x,\dots,x)}$ is clearly a positive multiple of ${x^d}$, and the claim now follows from (ii). $\Box$

If the variables ${x_1,\dots,x_n}$ are restricted to be nonnegative, the same argument gives Schur convexity for odd degrees also.

The proof in Hunter of positive definiteness is arranged a little differently than the one above, but still relies ultimately on the identity (2). I wonder if there is a genuinely different way to establish positive definiteness that does not go through this identity.

I’ve just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds“, submitted to Discrete and Continuous Dynamical Systems. This is a variant of my recent paper on the universality of potential well dynamics, but instead of trying to embed dynamical systems into a potential well ${\partial_{tt} u = -\nabla V(u)}$, here we try to embed dynamical systems into the incompressible Euler equations

$\displaystyle \partial_t u + \nabla_u u = - \mathrm{grad}_g p \ \ \ \ \ (1)$

$\displaystyle \mathrm{div}_g u = 0$

on a Riemannian manifold ${(M,g)}$. (One is particularly interested in the case of flat manifolds ${M}$, particularly ${{\bf R}^3}$ or ${({\bf R}/{\bf Z})^3}$, but for the main result of this paper it is essential that one is permitted to consider curved manifolds.) This system, first studied by Ebin and Marsden, is the natural generalisation of the usual incompressible Euler equations to curved space; it can be viewed as the formal geodesic flow equation on the infinite-dimensional manifold of volume-preserving diffeomorphisms on ${M}$ (see this previous post for a discussion of this in the flat space case).

The Euler equations can be viewed as a nonlinear equation in which the nonlinearity is a quadratic function of the velocity field ${u}$. It is thus natural to compare the Euler equations with quadratic ODE of the form

$\displaystyle \partial_t y = B(y,y) \ \ \ \ \ (2)$

where ${y: {\bf R} \rightarrow {\bf R}^n}$ is the unknown solution, and ${B: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}^n}$ is a bilinear map, which we may assume without loss of generality to be symmetric. One can ask whether such an ODE may be linearly embedded into the Euler equations on some Riemannian manifold ${(M,g)}$, which means that there is an injective linear map ${U: {\bf R}^n \rightarrow \Gamma(TM)}$ from ${{\bf R}^n}$ to smooth vector fields on ${M}$, as well as a bilinear map ${P: {\bf R}^n \times {\bf R}^n \rightarrow C^\infty(M)}$ to smooth scalar fields on ${M}$, such that the map ${y \mapsto (U(y), P(y,y))}$ takes solutions to (2) to solutions to (1), or equivalently that

$\displaystyle U(B(y,y)) + \nabla_{U(y)} U(y) = - \mathrm{grad}_g P(y,y)$

$\displaystyle \mathrm{div}_g U(y) = 0$

for all ${y \in {\bf R}^n}$.

For simplicity let us restrict ${M}$ to be compact. There is an obvious necessary condition for this embeddability to occur, which comes from energy conservation law for the Euler equations; unpacking everything, this implies that the bilinear form ${B}$ in (2) has to obey a cancellation condition

$\displaystyle \langle B(y,y), y \rangle = 0 \ \ \ \ \ (3)$

for some positive definite inner product ${\langle, \rangle: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}}$ on ${{\bf R}^n}$. The main result of the paper is the converse to this statement: if ${B}$ is a symmetric bilinear form obeying a cancellation condition (3), then it is possible to embed the equations (2) into the Euler equations (1) on some Riemannian manifold ${(M,g)}$; the catch is that this manifold will depend on the form ${B}$ and on the dimension ${n}$ (in fact in the construction I have, ${M}$ is given explicitly as ${SO(n) \times ({\bf R}/{\bf Z})^{n+1}}$, with a funny metric on it that depends on ${B}$).

As a consequence, any finite dimensional portion of the usual “dyadic shell models” used as simplified toy models of the Euler equation, can actually be embedded into a genuine Euler equation, albeit on a high-dimensional and curved manifold. This includes portions of the self-similar “machine” I used in a previous paper to establish finite time blowup for an averaged version of the Navier-Stokes (or Euler) equations. Unfortunately, the result in this paper does not apply to infinite-dimensional ODE, so I cannot yet establish finite time blowup for the Euler equations on a (well-chosen) manifold. It does not seem so far beyond the realm of possibility, though, that this could be done in the relatively near future. In particular, the result here suggests that one could construct something resembling a universal Turing machine within an Euler flow on a manifold, which was one ingredient I would need to engineer such a finite time blowup.

The proof of the main theorem proceeds by an “elimination of variables” strategy that was used in some of my previous papers in this area, though in this particular case the Nash embedding theorem (or variants thereof) are not required. The first step is to lessen the dependence on the metric ${g}$ by partially reformulating the Euler equations (1) in terms of the covelocity ${g \cdot u}$ (which is a ${1}$-form) instead of the velocity ${u}$. Using the freedom to modify the dimension of the underlying manifold ${M}$, one can also decouple the metric ${g}$ from the volume form that is used to obtain the divergence-free condition. At this point the metric can be eliminated, with a certain positive definiteness condition between the velocity and covelocity taking its place. After a substantial amount of trial and error (motivated by some “two-and-a-half-dimensional” reductions of the three-dimensional Euler equations, and also by playing around with a number of variants of the classic “separation of variables” strategy), I eventually found an ansatz for the velocity and covelocity that automatically solved most of the components of the Euler equations (as well as most of the positive definiteness requirements), as long as one could find a number of scalar fields that obeyed a certain nonlinear system of transport equations, and also obeyed a positive definiteness condition. Here I was stuck for a bit because the system I ended up with was overdetermined – more equations than unknowns. After trying a number of special cases I eventually found a solution to the transport system on the sphere, except that the scalar functions sometimes degenerated and so the positive definiteness property I wanted was only obeyed with positive semi-definiteness. I tried for some time to perturb this example into a strictly positive definite solution before eventually working out that this was not possible. Finally I had the brainwave to lift the solution from the sphere to an even more symmetric space, and this quickly led to the final solution of the problem, using the special orthogonal group rather than the sphere as the underlying domain. The solution ended up being rather simple in form, but it is still somewhat miraculous to me that it exists at all; in retrospect, given the overdetermined nature of the problem, relying on a large amount of symmetry to cut down the number of equations was basically the only hope.

I’ve just uploaded to the arXiv my paper “On the universality of potential well dynamics“, submitted to Dynamics of PDE. This is a spinoff from my previous paper on blowup of nonlinear wave equations, inspired by some conversations with Sungjin Oh. Here we focus mainly on the zero-dimensional case of such equations, namely the potential well equation

$\displaystyle \partial_{tt} u = - (\nabla F)(u) \ \ \ \ \ (1)$

for a particle ${u: {\bf R} \rightarrow {\bf R}^m}$ trapped in a potential well with potential ${F: {\bf R}^m \rightarrow {\bf R}}$, with ${F(z) \rightarrow +\infty}$ as ${z \rightarrow \infty}$. This ODE always admits global solutions from arbitrary initial positions ${u(0)}$ and initial velocities ${\partial_t u(0)}$, thanks to conservation of the Hamiltonian ${\frac{1}{2} |\partial_t u|^2 + F(u)}$. As this Hamiltonian is coercive (in that its level sets are compact), solutions to this equation are always almost periodic. On the other hand, as can already be seen using the harmonic oscillator ${\partial_{tt} u = - k^2 u}$ (and direct sums of this system), this equation can generate periodic solutions, as well as quasiperiodic solutions.

All quasiperiodic motions are almost periodic. However, there are many examples of dynamical systems that admit solutions that are almost periodic but not quasiperiodic. So one can pose the question: are the dynamics of potential wells universal in the sense that they can capture all almost periodic solutions?

A precise question can be phrased as follows. Let ${M}$ be a compact manifold, and let ${X}$ be a smooth vector field on ${M}$; to avoid degeneracies, let us take ${X}$ to be non-singular in the sense that it is everywhere non-vanishing. Then the trajectories of the first-order ODE

$\displaystyle \partial_t u = X(u) \ \ \ \ \ (2)$

for ${u: {\bf R} \rightarrow M}$ are always global and almost periodic. Can we then find a (coercive) potential ${F: {\bf R}^m \rightarrow {\bf R}}$ for some ${m}$, as well as a smooth embedding ${\phi: M \rightarrow {\bf R}^m}$, such that every solution ${u}$ to (2) pushes forward under ${\phi}$ to a solution to (1)? (Actually, for technical reasons it is preferable to map into the phase space ${{\bf R}^m \times {\bf R}^m}$, rather than position space ${{\bf R}^m}$, but let us ignore this detail for this discussion.)

It turns out that the answer is no; there is a very specific obstruction. Given a pair ${(M,X)}$ as above, define a strongly adapted ${1}$-form to be a ${1}$-form ${\phi}$ on ${M}$ such that ${\phi(X)}$ is pointwise positive, and the Lie derivative ${{\mathcal L}_X \phi}$ is an exact ${1}$-form. We then have

Theorem 1 A smooth compact non-singular dynamics ${(M,X)}$ can be embedded smoothly in a potential well system if and only if it admits a strongly adapted ${1}$-form.

For the “only if” direction, the key point is that potential wells (viewed as a Hamiltonian flow on the phase space ${{\bf R}^m \times {\bf R}^m}$) admit a strongly adapted ${1}$-form, namely the canonical ${1}$-form ${p dq}$, whose Lie derivative is the derivative ${dL}$ of the Lagrangian ${L := \frac{1}{2} |\partial_t u|^2 - F(u)}$ and is thus exact. The converse “if” direction is mainly a consequence of the Nash embedding theorem, and follows the arguments used in my previous paper.

Interestingly, the same obstruction also works for potential wells in a more general Riemannian manifold than ${{\bf R}^m}$, or for nonlinear wave equations with a potential; combining the two, the obstruction is also present for wave maps with a potential.

It is then natural to ask whether this obstruction is non-trivial, in the sense that there are at least some examples of dynamics ${(M,X)}$ that do not support strongly adapted ${1}$-forms (and hence cannot be modeled smoothly by the dynamics of a potential well, nonlinear wave equation, or wave maps). I posed this question on MathOverflow, and Robert Bryant provided a very nice construction, showing that the vector field ${(\sin(2\pi x), \cos(2\pi x))}$ on the ${2}$-torus ${({\bf R}/{\bf Z})^2}$ had no strongly adapted ${1}$-forms, and hence the dynamics of this vector field cannot be smoothly reproduced by a potential well, nonlinear wave equation, or wave map:

On the other hand, the suspension of any diffeomorphism does support a strongly adapted ${1}$-form (the derivative ${dt}$ of the time coordinate), and using this and the previous theorem I was able to embed a universal Turing machine into a potential well. In particular, there are flows for an explicitly describable potential well whose trajectories have behavior that is undecidable using the usual ZFC axioms of set theory! So potential well dynamics are “effectively” universal, despite the presence of the aforementioned obstruction.

In my previous work on blowup for Navier-Stokes like equations, I speculated that if one could somehow replicate a universal Turing machine within the Euler equations, one could use this machine to create a “von Neumann machine” that replicated smaller versions of itself, which on iteration would lead to a finite time blowup. Now that such a mechanism is present in nonlinear wave equations, it is tempting to try to make this scheme work in that setting. Of course, in my previous paper I had already demonstrated finite time blowup, at least in a three-dimensional setting, but that was a relatively simple discretely self-similar blowup in which no computation occurred. This more complicated blowup scheme would be significantly more effort to set up, but would be proof-of-concept that the same scheme would in principle be possible for the Navier-Stokes equations, assuming somehow that one can embed a universal Turing machine into the Euler equations. (But I’m still hopelessly stuck on how to accomplish this latter task…)