246B, Notes 1: Zeroes, poles, and factorisation of meromorphic functions

Previous set of notes: 246A Notes 5. Next set of notes: Notes 2.

— 1. Jensen’s formula —

Suppose is a non-zero rational function , then by the fundamental theorem of algebra one can write

for some non-zero constant , where ranges over the zeroes of (counting multiplicity) and ranges over the zeroes of (counting multiplicity), and assuming avoids the zeroes of . Taking absolute values and then logarithms, we arrive at the formula

as long as avoids the zeroes of both and . (In this set of notes we use for the natural logarithm when applied to a positive real number, and for the standard branch of the complex logarithm (which extends ); the multi-valued complex logarithm will only be used in passing.) Alternatively, taking logarithmic derivatives, we arrive at the closely related formula

again for avoiding the zeroes of both and . Thus we see that the zeroes and poles of a rational function describe the behaviour of that rational function, as well as close relatives of that function such as the log-magnitude and log-derivative . We have already seen these sorts of formulae arise in our treatment of the argument principle in 246A Notes 4.

Exercise 1 Let be a complex polynomial of degree .

• (i) (Gauss-Lucas theorem) Show that the complex roots of are contained in the closed convex hull of the complex roots of .
• (ii) (Laguerre separation theorem) If all the complex roots of are contained in a disk , and , then all the complex roots of are also contained in . (Hint: apply a suitable Möbius transformation to move to infinity, and then apply part (i) to a polynomial that emerges after applying this transformation.)

There are a number of useful ways to extend these formulae to more general meromorphic functions than rational functions. Firstly there is a very handy “local” variant of (1) known as Jensen’s formula:

Theorem 2 (Jensen’s formula) Let be a meromorphic function on an open neighbourhood of a disk , with all removable singularities removed. Then, if is neither a zero nor a pole of , we have

where and range over the zeroes and poles of respectively (counting multiplicity) in the disk .

One can view (3) as a truncated (or localised) variant of (1). Note also that the summands are always non-positive.

Proof: By perturbing slightly if necessary, we may assume that none of the zeroes or poles of (which form a discrete set) lie on the boundary circle . By translating and rescaling, we may then normalise and , thus our task is now to show that

We may remove the poles and zeroes inside the disk by the useful device of Blaschke products. Suppose for instance that has a zero inside the disk . Observe that the function

has magnitude on the unit circle , equals at the origin, has a simple zero at , but has no other zeroes or poles inside the disk. Thus Jensen’s formula (4) already holds if is replaced by . To prove (4) for , it thus suffices to prove it for , which effectively deletes a zero inside the disk from (and replaces it instead with its inversion ). Similarly we may remove all the poles inside the disk. As a meromorphic function only has finitely many poles and zeroes inside a compact set, we may thus reduce to the case when has no poles or zeroes on or inside the disk , at which point our goal is simply to show that

Since has no zeroes or poles inside the disk, it has a holomorphic logarithm (Exercise 46 of 246A Notes 4). In particular, is the real part of . The claim now follows by applying the mean value property (Exercise 17 of 246A Notes 3) to .

An important special case of Jensen’s formula arises when is holomorphic in a neighborhood of , in which case there are no contributions from poles and one simply has

This is quite a useful formula, mainly because the summands are non-negative; it can be viewed as a more precise assertion of the subharmonicity of (see Exercises 60(ix) and 61 of 246A Notes 5). Here are some quick applications of this formula:

Exercise 3 Use (6) to give another proof of Liouville’s theorem: a bounded holomorphic function on the entire complex plane is necessarily constant.

Exercise 4 Use Jensen’s formula to prove the fundamental theorem of algebra: a complex polynomial of degree has exactly complex zeroes (counting multiplicity), and can thus be factored as for some complex numbers with . (Note that the fundamental theorem was invoked previously in this section, but only for motivational purposes, so the proof here is non-circular.)

Exercise 5 (Shifted Jensen’s formula) Let be a meromorphic function on an open neighbourhood of a disk , with all removable singularities removed. Show that

for all in the open disk that are not zeroes or poles of , where and . (The function appearing in the integrand is sometimes known as the Poisson kernel, particularly if one normalises so that and .)

Exercise 6 (Bounded type)

• (i) If is a holomorphic function on that is not identically zero, show that .
• (ii) If is a meromorphic function on that is the ratio of two bounded holomorphic functions that are not identically zero, show that . (Functions of this form are said to be of bounded type and lie in the Nevanlinna class for the unit disk .)

Exercise 7 (Smoothed out Jensen formula) Let be a meromorphic function on an open set , and let be a smooth compactly supported function. Show that

where range over the zeroes and poles of (respectively) in the support of . Informally argue why this identity is consistent with Jensen’s formula.

When applied to entire functions , Jensen’s formula relates the order of growth of near infinity with the density of zeroes of . Here is a typical result:

Proposition 8 Let be an entire function, not identically zero, that obeys a growth bound for some and all . Then there exists a constant such that has at most zeroes (counting multiplicity) for any .

Entire functions that obey a growth bound of the form for every and (where depends on ) are said to be of order at most . The above theorem shows that for such functions that are not identically zero, the number of zeroes in a disk of radius does not grow much faster than . This is often a useful preliminary upper bound on the zeroes of entire functions, as the order of an entire function tends to be relatively easy to compute in practice.

Proof: First suppose that is non-zero. From (6) applied with and one has

Every zero in contribute at least to a summand on the right-hand side, while all other zeroes contribute a non-negative quantity, thus

where denotes the number of zeroes in . This gives the claim for . When , one can shift by a small amount to make non-zero at the origin (using the fact that zeroes of holomorphic functions not identically zero are isolated), modifying in the process, and then repeating the previous arguments.

Just as (3) and (7) give truncated variants of (1), we can create truncated versions of (2). The following crude truncation is adequate for many applications:

Theorem 9 (Truncated formula for log-derivative) Let be a holomorphic function on an open neighbourhood of a disk that is not identically zero on this disk. Suppose that one has a bound of the form for some and all on the circle . Let be constants. Then one has the approximate formula

for all in the disk other than zeroes of . Furthermore, the number of zeroes in the above sum is .

Proof: To abbreviate notation, we allow all implied constants in this proof to depend on .

We mimic the proof of Jensen’s formula. Firstly, we may translate and rescale so that and , so we have when , and our main task is to show that

for . Note that if then vanishes on the unit circle and hence (by the maximum principle) vanishes identically on the disk, a contradiction, so we may assume . From hypothesis we then have

on the unit circle, and so from Jensen’s formula (3) we see that

In particular we see that the number of zeroes with is , as claimed.

Suppose has a zero with . If we factor , where is the Blaschke product (5), then

Observe from Taylor expansion that the distance between and is , and hence for . Thus we see from (9) that we may use Blaschke products to remove all the zeroes in the annulus while only affecting the left-hand side of (8) by ; also, removing the Blaschke products does not affect on the unit circle, and only affects by thanks to (9). Thus we may assume without loss of generality that there are no zeroes in this annulus.

Similarly, given a zero with , we have , so using Blaschke products to remove all of these zeroes also only affects the left-hand side of (8) by (since the number of zeroes here is ), with also modified by at most . Thus we may assume in fact that has no zeroes whatsoever within the unit disk. We may then also normalise , then for all . By Jensen’s formula again, we have

and thus (by using the identity for any real )

On the other hand, from (7) we have

which implies from (10) that and its first derivatives are on the disk . But recall from the proof of Jensen’s formula that is the derivative of a logarithm of , whose real part is . By the Cauchy-Riemann equations for , we conclude that on the disk , as required.

Exercise 10

• (i) (Borel-Carathéodory theorem) If is analytic on an open neighborhood of a disk , show that

  

  (Hint: one can normalise , , , and . Now maps the unit disk to the half-plane . Use a Möbius transformation to map the half-plane to the unit disk and then use the Schwarz lemma.)
• (ii) Use (i) to give an alternate way to conclude the proof of Theorem 9.

A variant of the above argument allows one to make precise the heuristic that holomorphic functions locally look like polynomials:

Exercise 11 (Local Weierstrass factorisation) Let the notation and hypotheses be as in Theorem 9. Then show that

for all in the disk , where is a polynomial whose zeroes are precisely the zeroes of in (counting multiplicity) and is a holomorphic function on of magnitude and first derivative on this disk. Furthermore, show that the degree of is .

Exercise 12 (Preliminary Beurling factorisation) Let denote the space of bounded analytic functions on the unit disk; this is a normed vector space with norm

• (i) If is not identically zero, and denote the zeroes of in counting multiplicity, show that

  

  and

  

• (ii) Let the notation be as in (i). If we define the Blaschke product

  

  where is the order of vanishing of at zero, show that this product converges absolutely to a holomorphic function on , and that for all . (It may be easier to work with finite Blaschke products first to obtain this bound.)
• (iii) Continuing the notation from (i), establish a factorisation for some holomorphic function with for all .
• (iv) (Theorem of F. and M. Riesz, special case) If extends continuously to the boundary , show that the set has zero measure.

Remark 13 The factorisation (iii) can be refined further, with being the Poisson integral of some finite measure on the unit circle. Using the Lebesgue decomposition of this finite measure into absolutely continuous parts one ends up factorising functions into “outer functions” and “inner functions”, giving the Beurling factorisation of . There are also extensions to larger spaces than (which are to as is to ), known as Hardy spaces. We will not discuss this topic further here, but see for instance this text of Garnett for a treatment.

Exercise 14 (Littlewood’s lemma) Let be holomorphic on an open neighbourhood of a rectangle for some and , with non-vanishing on the boundary of the rectangle. Show that

where ranges over the zeroes of inside (counting multiplicity) and one uses a branch of which is continuous on the upper, lower, and right edges of . (This lemma is a popular tool to explore the zeroes of Dirichlet series such as the Riemann zeta function.)

Course announcement: 246B, complex analysis

Just a short announcement that next quarter I will be continuing the recently concluded 246A complex analysis class as 246B. Topics I plan to cover:

• Schwartz-Christoffel transformations and the uniformisation theorem (using the remainder of the 246A notes);
• Jensen’s formula and factorisation theorems (particularly Weierstrass and Hadamard); the Gamma function;
• Connections with the Fourier transform on the real line;
• Elliptic functions and their relatives;
• (if time permits) the Riemann zeta function and the prime number theorem.

Notes for the later material will appear on this blog in due course.

Sendov’s conjecture for sufficiently high degree polynomials

I’ve just uploaded to the arXiv my paper “Sendov’s conjecture for sufficiently high degree polynomials“. This paper is a contribution to an old conjecture of Sendov on the zeroes of polynomials:

Conjecture 1 (Sendov’s conjecture) Let be a polynomial of degree that has all zeroes in the closed unit disk . If is one of these zeroes, then has at least one zero in .

It is common in the literature on this problem to normalise to be monic, and to rotate the zero to be an element of the unit interval . As it turns out, the location of on this unit interval ends up playing an important role in the arguments.

Many cases of this conjecture are already known, for instance

• When (Brown-Xiang 1999);
• When (Gauss-Lucas theorem);
• When (Bojanov 2011);
• When for a fixed , and is sufficiently large depending on (Dégot 2014);
• When for a sufficiently large absolute constant (Chalebgwa 2020);
• When (Rubinstein 1968; Goodman-Rahman-Ratti 1969; Joyal 1969);
• When , where is sufficiently small depending on (Miller 1993; Vajaitu-Zaharescu 1993);
• When (Chijiwa 2011);
• When (Kasmalkar 2014).

In particular, in high degrees the only cases left uncovered by prior results are when is close (but not too close) to , or when is close (but not too close) to ; see Figure 1 of my paper.

Our main result covers the high degree case uniformly for all values of :

Theorem 2 There exists an absolute constant such that Sendov’s conjecture holds for all .

In principle, this reduces the verification of Sendov’s conjecture to a finite time computation, although our arguments use compactness methods and thus do not easily provide an explicit value of . I believe that the compactness arguments can be replaced with quantitative substitutes that provide an explicit , but the value of produced is likely to be extremely large (certainly much larger than ).

Because of the previous results (particularly those of Chalebgwa and Chijiwa), we will only need to establish the following two subcases of the above theorem:

Theorem 3 (Sendov’s conjecture near the origin) Under the additional hypothesis , Sendov’s conjecture holds for sufficiently large .

Theorem 4 (Sendov’s conjecture near the unit circle) Under the additional hypothesis for a fixed , Sendov’s conjecture holds for sufficiently large .

We approach these theorems using the “compactness and contradiction” strategy, assuming that there is a sequence of counterexamples whose degrees going to infinity, using various compactness theorems to extract various asymptotic objects in the limit , and somehow using these objects to derive a contradiction. There are many ways to effect such a strategy; we will use a formalism that I call “cheap nonstandard analysis” and which is common in the PDE literature, in which one repeatedly passes to subsequences as necessary whenever one invokes a compactness theorem to create a limit object. However, the particular choice of asymptotic formalism one selects is not of essential importance for the arguments.

I also found it useful to use the language of probability theory. Given a putative counterexample to Sendov’s conjecture, let be a zero of (chosen uniformly at random among the zeroes of , counting multiplicity), and let similarly be a uniformly random zero of . We introduce the logarithmic potentials

and the Stieltjes transforms

Standard calculations using the fundamental theorem of algebra yield the basic identities

and

and in particular the random variables are linked to each other by the identity

On the other hand, the hypotheses of Sendov’s conjecture (and the Gauss-Lucas theorem) place inside the unit disk . Applying Prokhorov’s theorem, and passing to a subsequence, one can then assume that the random variables converge in distribution to some limiting random variables (possibly defined on a different probability space than the original variables ), also living almost surely inside the unit disk. Standard potential theory then gives the convergence

and

at least in the local sense. Among other things, we then conclude from the identity (2) and some elementary inequalities that

for all . This turns out to have an appealing interpretation in terms of Brownian motion: if one takes two Brownian motions in the complex plane, one originating from and one originating from , then the location where these Brownian motions first exit the unit disk will have the same distribution. (In our paper we actually replace Brownian motion with the closely related formalism of balayage.) This turns out to connect the random variables , quite closely to each other. In particular, with this observation and some additional arguments involving both the unique continuation property for harmonic functions and Grace’s theorem (discussed in this previous post), with the latter drawn from the prior work of Dégot, we can get very good control on these distributions:

Theorem 5

• (i) If , then almost surely lie in the semicircle and have the same distribution.
• (ii) If , then is uniformly distributed on the circle , and is almost surely zero.

In case (i) (and strengthening the hypothesis to to control some technical contributions of “outlier” zeroes of ), we can use this information about and (4) to ensure that the normalised logarithmic derivative has a non-negative winding number in a certain small (but not too small) circle around the origin, which by the argument principle is inconsistent with the hypothesis that has a zero at and that has no zeroes near . This is how we establish Theorem 3.

Case (ii) turns out to be more delicate. This is because there are a number of “near-counterexamples” to Sendov’s conjecture that are compatible with the hypotheses and conclusion of case (ii). The simplest such example is , where the zeroes of are uniformly distributed amongst the roots of unity (including at ), and the zeroes of are all located at the origin. In my paper I also discuss a variant of this construction, in which has zeroes mostly near the origin, but also acquires a bounded number of zeroes at various locations inside the unit disk. Specifically, we take

where for some constants and

By a perturbative analysis to locate the zeroes of , one eventually would be able to arrive at a true counterexample to Sendov’s conjecture if these locations were in the open lune

and if one had the inequality

for all . However, if one takes the mean of this inequality in , one arrives at the inequality

which is incompatible with the hypotheses and . In order to extend this argument to more general polynomials , we require a stability analysis of the endpoint equation

where we now only assume the closed conditions and . The above discussion then places all the zeros on the arc

and if one also takes the second Fourier coefficient of (6) one also obtains the vanishing second moment

These two conditions are incompatible with each other (except in the degenerate case when all the vanish), because all the non-zero elements of the arc (7) have argument in , so in particular their square will have negative real part. It turns out that one can adapt this argument to the more general potential counterexamples to Sendov’s conjecture (in the form of Theorem 4). The starting point is to use (1), (4), and Theorem 5(ii) to obtain good control on , which one then integrates and exponentiates to get good control on , and then on a second integration one gets enough information about to pin down the location of its zeroes to high accuracy. The constraint that these zeroes lie inside the unit disk then gives an inequality resembling (5), and an adaptation of the above stability analysis is then enough to conclude. The arguments here are inspired by the previous arguments of Miller, which treated the case when was extremely close to via a similar perturbative analysis; the main novelty is to control the error terms not in terms of the magnitude of the largest zero of (which is difficult to manage when gets large), but rather by the variance of those zeroes, which ends up being a more tractable expression to keep track of.

Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle

Laura Cladek and I have just uploaded to the arXiv our paper “Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle“. This paper concerns a continuous version of the notion of additive energy. Given a finite measure on and a scale , define the energy at scale to be the quantity

where is the product measure on formed from four copies of the measure on . We will be interested in Cantor-type measures , supported on a compact set and obeying the Ahlfors-David regularity condition

for all balls and some constants , as well as the matching lower bound

when whenever . One should think of as a -dimensional fractal set, and as some vaguely self-similar measure on this set.

Note that once one fixes , the variable in (1) is constrained to a ball of radius , hence we obtain the trivial upper bound

If the set contains a lot of “additive structure”, one can expect this bound to be basically sharp; for instance, if is an integer, is a -dimensional unit disk, and is Lebesgue measure on this disk, one can verify that (where we allow implied constants to depend on . However we show that if the dimension is non-integer, then one obtains a gain:

Theorem 1 If is not an integer, and are as above, then

for some depending only on .

Informally, this asserts that Ahlfors-David regular fractal sets of non-integer dimension cannot behave as if they are approximately closed under addition. In fact the gain we obtain is quasipolynomial in the regularity constant :

(We also obtain a localised version in which the regularity condition is only required to hold at scales between and .) Such a result was previously obtained (with more explicit values of the implied constants) in the one-dimensional case by Dyatlov and Zahl; but in higher dimensions there does not appear to have been any results for this general class of sets and measures . In the paper of Dyatlov and Zahl it is noted that some dependence on is necessary; in particular, cannot be much better than . This reflects the fact that there are fractal sets that do behave reasonably well with respect to addition (basically because they are built out of long arithmetic progressions at many scales); however, such sets are not very Ahlfors-David regular. Among other things, this result readily implies a dimension expansion result

for any non-degenerate smooth map , including the sum map and (in one dimension) the product map , where the non-degeneracy condition required is that the gradients are invertible for every . We refer to the paper for the formal statement.

Our higher-dimensional argument shares many features in common with that of Dyatlov and Zahl, notably a reliance on the modern tools of additive combinatorics (and specifically the Bogulybov-Ruzsa lemma of Sanders). However, in one dimension we were also able to find a completely elementary argument, avoiding any particularly advanced additive combinatorics and instead primarily exploiting the order-theoretic properties of the real line, that gave a superior value of , namely

One of the main reasons for obtaining such improved energy bounds is that they imply a fractal uncertainty principle in some regimes. We focus attention on the model case of obtaining such an uncertainty principle for the semiclassical Fourier transform

where is a small parameter. If are as above, and denotes the -neighbourhood of , then from the Hausdorff-Young inequality one obtains the trivial bound

(There are also variants involving pairs of sets , but for simplicity we focus on the uncertainty principle for a single set .) The fractal uncertainty principle, when it applies, asserts that one can improve this to

for some ; informally, this asserts that a function and its Fourier transform cannot simultaneously be concentrated in the set when , and that a function cannot be concentrated on and have its Fourier transform be of maximum size on when . A modification of the disk example mentioned previously shows that such a fractal uncertainty principle cannot hold if is an integer. However, in one dimension, the fractal uncertainty principle is known to hold for all . The above-mentioned results of Dyatlov and Zahl were able to establish this for close to , and the remaining cases and were later established by Bourgain-Dyatlov and Dyatlov-Jin respectively. Such uncertainty principles have applications to hyperbolic dynamics, in particular in establishing spectral gaps for certain Selberg zeta functions.

It remains a largely open problem to establish a fractal uncertainty principle in higher dimensions. Our results allow one to establish such a principle when the dimension is close to , and is assumed to be odd (to make a non-integer). There is also work of Han and Schlag that obtains such a principle when one of the copies of is assumed to have a product structure. We hope to obtain further higher-dimensional fractal uncertainty principles in subsequent work.

We now sketch how our main theorem is proved. In both one dimension and higher dimensions, the main point is to get a preliminary improvement

over the trivial bound (2) for any small , provided is sufficiently small depending on ; one can then iterate this bound by a fairly standard “induction on scales” argument (which roughly speaking can be used to show that energies behave somewhat multiplicatively in the scale parameter ) to propagate the bound to a power gain at smaller scales. We found that a particularly clean way to run the induction on scales was via use of the Gowers uniformity norm , and particularly via a clean Fubini-type inequality

(ultimately proven using the Gowers-Cauchy-Schwarz inequality) that allows one to “decouple” coarse and fine scale aspects of the Gowers norms (and hence of additive energies).

It remains to obtain the preliminary improvement. In one dimension this is done by identifying some “left edges” of the set that supports : intervals that intersect , but such that a large interval just to the left of this interval is disjoint from . Here is a large constant and is a scale parameter. It is not difficult to show (using in particular the Archimedean nature of the real line) that if one has the Ahlfors-David regularity condition for some then left edges exist in abundance at every scale; for instance most points of would be expected to lie in quite a few of these left edges (much as most elements of, say, the ternary Cantor set would be expected to contain a lot of s in their base expansion). In particular, most pairs would be expected to lie in a pair of left edges of equal length. The key point is then that if lies in such a pair with , then there are relatively few pairs at distance from for which one has the relation , because will both tend to be to the right of respectively. This causes a decrement in the energy at scale , and by carefully combining all these energy decrements one can eventually cobble together the energy bound (3).

We were not able to make this argument work in higher dimension (though perhaps the cases and might not be completely out of reach from these methods). Instead we return to additive combinatorics methods. If the claim (3) failed, then by applying the Balog-Szemeredi-Gowers theorem we can show that the set has high correlation with an approximate group , and hence (by the aforementioned Bogulybov-Ruzsa type theorem of Sanders, which is the main source of the quasipolynomial bounds in our final exponent) will exhibit an approximate “symmetry” along some non-trivial arithmetic progression of some spacing length and some diameter . The -neighbourhood of will then resemble the union of parallel “cylinders” of dimensions . If we focus on a typical -ball of , the set now resembles a Cartesian product of an interval of length with a subset of a -dimensional hyperplane, which behaves approximately like an Ahlfors-David regular set of dimension (this already lets us conclude a contradiction if ). Note that if the original dimension was non-integer then this new dimension will also be non-integer. It is then possible to contradict the failure of (3) by appealing to a suitable induction hypothesis at one lower dimension.