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I recently came across this question on MathOverflow asking if there are any polynomials of two variables with rational coefficients, such that the map is a bijection. The answer to this question is almost surely “no”, but it is remarkable how hard this problem resists any attempt at rigorous proof. (MathOverflow users with enough privileges to see deleted answers will find that there are no fewer than seventeen deleted attempts at a proof in response to this question!)

On the other hand, the one surviving response to the question does point out this paper of Poonen which shows that assuming a powerful conjecture in Diophantine geometry known as the Bombieri-Lang conjecture (discussed in this previous post), it is at least possible to exhibit polynomials which are injective.

I believe that it should be possible to also rule out the existence of bijective polynomials if one assumes the Bombieri-Lang conjecture, and have sketched out a strategy to do so, but filling in the gaps requires a fair bit more algebraic geometry than I am capable of. So as a sort of experiment, I would like to see if a rigorous implication of this form (similarly to the rigorous implication of the Erdos-Ulam conjecture from the Bombieri-Lang conjecture in my previous post) can be crowdsourced, in the spirit of the polymath projects (though I feel that this particular problem should be significantly quicker to resolve than a typical such project).

Here is how I imagine a Bombieri-Lang-powered resolution of this question should proceed (modulo a large number of unjustified and somewhat vague steps that I believe to be true but have not established rigorously). Suppose for contradiction that we have a bijective polynomial . Then for any polynomial of one variable, the surface

has infinitely many rational points; indeed, every rational lifts to exactly one rational point in . I believe that for “typical” this surface should be irreducible. One can now split into two cases:

- (a) The rational points in are Zariski dense in .
- (b) The rational points in are not Zariski dense in .

Consider case (b) first. By definition, this case asserts that the rational points in are contained in a finite number of algebraic curves. By Faltings’ theorem (a special case of the Bombieri-Lang conjecture), any curve of genus two or higher only contains a finite number of rational points. So all but finitely many of the rational points in are contained in a finite union of genus zero and genus one curves. I think all genus zero curves are birational to a line, and all the genus one curves are birational to an elliptic curve (though I don’t have an immediate reference for this). These curves all can have an infinity of rational points, but very few of them should have “enough” rational points that their projection to the third coordinate is “large”. In particular, I believe

- (i) If is birational to an elliptic curve, then the number of elements of of height at most should grow at most polylogarithmically in (i.e., be of order .
- (ii) If is birational to a line but not of the form for some rational , then then the number of elements of of height at most should grow slower than (in fact I think it can only grow like ).

I do not have proofs of these results (though I think something similar to (i) can be found in Knapp’s book, and (ii) should basically follow by using a rational parameterisation of with nonlinear). Assuming these assertions, this would mean that there is a curve of the form that captures a “positive fraction” of the rational points of , as measured by restricting the height of the third coordinate to lie below a large threshold , computing density, and sending to infinity (taking a limit superior). I believe this forces an identity of the form

for all . Such identities are certainly possible for some choices of (e.g. for arbitrary polynomials of one variable) but I believe that the only way that such identities hold for a “positive fraction” of (as measured using height as before) is if there is in fact a rational identity of the form

for some rational functions with rational coefficients (in which case we would have and ). But such an identity would contradict the hypothesis that is bijective, since one can take a rational point outside of the curve , and set , in which case we have violating the injective nature of . Thus, modulo a lot of steps that have not been fully justified, we have ruled out the scenario in which case (b) holds for a “positive fraction” of .

This leaves the scenario in which case (a) holds for a “positive fraction” of . Assuming the Bombieri-Lang conjecture, this implies that for such , any resolution of singularities of fails to be of general type. I would imagine that this places some very strong constraints on , since I would expect the equation to describe a surface of general type for “generic” choices of (after resolving singularities). However, I do not have a good set of techniques for detecting whether a given surface is of general type or not. Presumably one should proceed by viewing the surface as a fibre product of the simpler surface and the curve over the line . In any event, I believe the way to handle (a) is to show that the failure of general type of implies some strong algebraic constraint between and (something in the spirit of (1), perhaps), and then use this constraint to rule out the bijectivity of by some further *ad hoc* method.

(This post is mostly intended for my own reference, as I found myself repeatedly looking up several conversions between polynomial bases on various occasions.)

Let denote the vector space of polynomials of one variable with real coefficients of degree at most . This is a vector space of dimension , and the sequence of these spaces form a filtration:

A standard basis for these vector spaces are given by the monomials : every polynomial in can be expressed uniquely as a linear combination of the first monomials . More generally, if one has any sequence of polynomials, with each of degree exactly , then an easy induction shows that forms a basis for .

In particular, if we have *two* such sequences and of polynomials, with each of degree and each of degree , then must be expressible uniquely as a linear combination of the polynomials , thus we have an identity of the form

for some *change of basis coefficients* . These coefficients describe how to convert a polynomial expressed in the basis into a polynomial expressed in the basis.

Many standard combinatorial quantities involving two natural numbers can be interpreted as such change of basis coefficients. The most familiar example are the binomial coefficients , which measures the conversion from the shifted monomial basis to the monomial basis , thanks to (a special case of) the binomial formula:

thus for instance

More generally, for any shift , the conversion from to is measured by the coefficients , thanks to the general case of the binomial formula.

But there are other bases of interest too. For instance if one uses the falling factorial basis

then the conversion from falling factorials to monomials is given by the Stirling numbers of the first kind :

thus for instance

and the conversion back is given by the Stirling numbers of the second kind :

thus for instance

If one uses the binomial functions as a basis instead of the falling factorials, one of course can rewrite these conversions as

and

thus for instance

and

As a slight variant, if one instead uses rising factorials

then the conversion to monomials yields the unsigned Stirling numbers of the first kind:

thus for instance

One final basis comes from the polylogarithm functions

For instance one has

and more generally one has

for all natural numbers and some polynomial of degree (the *Eulerian polynomials*), which when converted to the monomial basis yields the (shifted) Eulerian numbers

For instance

These particular coefficients also have useful combinatorial interpretations. For instance:

- The binomial coefficient is of course the number of -element subsets of .
- The unsigned Stirling numbers of the first kind are the number of permutations of with exactly cycles. The signed Stirling numbers are then given by the formula .
- The Stirling numbers of the second kind are the number of ways to partition into non-empty subsets.
- The Eulerian numbers are the number of permutations of with exactly ascents.

These coefficients behave similarly to each other in several ways. For instance, the binomial coefficients obey the well known Pascal identity

(with the convention that vanishes outside of the range ). In a similar spirit, the unsigned Stirling numbers of the first kind obey the identity

and the signed counterparts obey the identity

The Stirling numbers of the second kind obey the identity

and the Eulerian numbers obey the identity

This is a sequel to this previous blog post, in which we discussed the effect of the heat flow evolution

on the zeroes of a time-dependent family of polynomials , with a particular focus on the case when the polynomials had real zeroes. Here (inspired by some discussions I had during a recent conference on the Riemann hypothesis in Bristol) we record the analogous theory in which the polynomials instead have zeroes on a circle , with the heat flow slightly adjusted to compensate for this. As we shall discuss shortly, a key example of this situation arises when is the numerator of the zeta function of a curve.

More precisely, let be a natural number. We will say that a polynomial

of degree (so that ) obeys the *functional equation* if the are all real and

for all , thus

and

for all non-zero . This means that the zeroes of (counting multiplicity) lie in and are symmetric with respect to complex conjugation and inversion across the circle . We say that this polynomial *obeys the Riemann hypothesis* if all of its zeroes actually lie on the circle . For instance, in the case, the polynomial obeys the Riemann hypothesis if and only if .

Such polynomials arise in number theory as follows: if is a projective curve of genus over a finite field , then, as famously proven by Weil, the associated local zeta function (as defined for instance in this previous blog post) is known to take the form

where is a degree polynomial obeying both the functional equation and the Riemann hypothesis. In the case that is an elliptic curve, then and takes the form , where is the number of -points of minus . The Riemann hypothesis in this case is a famous result of Hasse.

Another key example of such polynomials arise from rescaled characteristic polynomials

of matrices in the compact symplectic group . These polynomials obey both the functional equation and the Riemann hypothesis. The Sato-Tate conjecture (in higher genus) asserts, roughly speaking, that “typical” polyomials arising from the number theoretic situation above are distributed like the rescaled characteristic polynomials (1), where is drawn uniformly from with Haar measure.

Given a polynomial of degree with coefficients

we can evolve it in time by the formula

thus for . Informally, as one increases , this evolution accentuates the effect of the extreme monomials, particularly, and at the expense of the intermediate monomials such as , and conversely as one decreases . This family of polynomials obeys the heat-type equation

In view of the results of Marcus, Spielman, and Srivastava, it is also very likely that one can interpret this flow in terms of expected characteristic polynomials involving conjugation over the compact symplectic group , and should also be tied to some sort of “” version of Brownian motion on this group, but we have not attempted to work this connection out in detail.

It is clear that if obeys the functional equation, then so does for any other time . Now we investigate the evolution of the zeroes. Suppose at some time that the zeroes of are distinct, then

From the inverse function theorem we see that for times sufficiently close to , the zeroes of continue to be distinct (and vary smoothly in ), with

Differentiating this at any not equal to any of the , we obtain

and

and

Inserting these formulae into (2) (expanding as ) and canceling some terms, we conclude that

for sufficiently close to , and not equal to . Extracting the residue at , we conclude that

which we can rearrange as

If we make the change of variables (noting that one can make depend smoothly on for sufficiently close to ), this becomes

Intuitively, this equation asserts that the phases repel each other if they are real (and attract each other if their difference is imaginary). If obeys the Riemann hypothesis, then the are all real at time , then the Picard uniqueness theorem (applied to and its complex conjugate) then shows that the are also real for sufficiently close to . If we then define the entropy functional

then the above equation becomes a gradient flow

which implies in particular that is non-increasing in time. This shows that as one evolves time forward from , there is a uniform lower bound on the separation between the phases , and hence the equation can be solved indefinitely; in particular, obeys the Riemann hypothesis for all if it does so at time . Our argument here assumed that the zeroes of were simple, but this assumption can be removed by the usual limiting argument.

For any polynomial obeying the functional equation, the rescaled polynomials converge locally uniformly to as . By Rouche’s theorem, we conclude that the zeroes of converge to the equally spaced points on the circle . Together with the symmetry properties of the zeroes, this implies in particular that obeys the Riemann hypothesis for all sufficiently large positive . In the opposite direction, when , the polynomials converge locally uniformly to , so if , of the zeroes converge to the origin and the other converge to infinity. In particular, fails the Riemann hypothesis for sufficiently large negative . Thus (if ), there must exist a real number , which we call the *de Bruijn-Newman constant* of the original polynomial , such that obeys the Riemann hypothesis for and fails the Riemann hypothesis for . The situation is a bit more complicated if vanishes; if is the first natural number such that (or equivalently, ) does not vanish, then by the above arguments one finds in the limit that of the zeroes go to the origin, go to infinity, and the remaining zeroes converge to the equally spaced points . In this case the de Bruijn-Newman constant remains finite except in the degenerate case , in which case .

For instance, consider the case when and for some real with . Then the quadratic polynomial

has zeroes

and one easily checks that these zeroes lie on the circle when , and are on the real axis otherwise. Thus in this case we have (with if ). Note how as increases to , the zeroes repel each other and eventually converge to , while as decreases to , the zeroes collide and then separate on the real axis, with one zero going to the origin and the other to infinity.

The arguments in my paper with Brad Rodgers (discussed in this previous post) indicate that for a “typical” polynomial of degree that obeys the Riemann hypothesis, the expected time to relaxation to equilibrium (in which the zeroes are equally spaced) should be comparable to , basically because the average spacing is and hence by (3) the typical velocity of the zeroes should be comparable to , and the diameter of the unit circle is comparable to , thus requiring time comparable to to reach equilibrium. Taking contrapositives, this suggests that the de Bruijn-Newman constant should typically take on values comparable to (since typically one would not expect the initial configuration of zeroes to be close to evenly spaced). I have not attempted to formalise or prove this claim, but presumably one could do some numerics (perhaps using some of the examples of given previously) to explore this further.

Let be a monic polynomial of degree with complex coefficients. Then by the fundamental theorem of algebra, we can factor as

for some complex zeroes (possibly with repetition).

Now suppose we evolve with respect to time by heat flow, creating a function of two variables with given initial data for which

On the space of polynomials of degree at most , the operator is nilpotent, and one can solve this equation explicitly both forwards and backwards in time by the Taylor series

For instance, if one starts with a quadratic , then the polynomial evolves by the formula

As the polynomial evolves in time, the zeroes evolve also. Assuming for sake of discussion that the zeroes are simple, the inverse function theorem tells us that the zeroes will (locally, at least) evolve smoothly in time. What are the dynamics of this evolution?

For instance, in the quadratic case, the quadratic formula tells us that the zeroes are

and

after arbitrarily choosing a branch of the square root. If are real and the discriminant is initially positive, we see that we start with two real zeroes centred around , which then approach each other until time , at which point the roots collide and then move off from each other in an imaginary direction.

In the general case, we can obtain the equations of motion by implicitly differentiating the defining equation

in time using (2) to obtain

To simplify notation we drop the explicit dependence on time, thus

From (1) and the product rule, we see that

and

(where all indices are understood to range over ) leading to the equations of motion

at least when one avoids those times in which there is a repeated zero. In the case when the zeroes are real, each term represents a (first-order) attraction in the dynamics between and , but the dynamics are more complicated for complex zeroes (e.g. purely imaginary zeroes will experience repulsion rather than attraction, as one already sees in the quadratic example). Curiously, this system resembles that of Dyson brownian motion (except with the brownian motion part removed, and time reversed). I learned of the connection between the ODE (3) and the heat equation from this paper of Csordas, Smith, and Varga, but perhaps it has been mentioned in earlier literature as well.

One interesting consequence of these equations is that if the zeroes are real at some time, then they will stay real as long as the zeroes do not collide. Let us now restrict attention to the case of real simple zeroes, in which case we will rename the zeroes as instead of , and order them as . The evolution

can now be thought of as reverse gradient flow for the “entropy”

(which is also essentially the logarithm of the discriminant of the polynomial) since we have

In particular, we have the monotonicity formula

where is the “energy”

where in the last line we use the antisymmetrisation identity

Among other things, this shows that as one goes backwards in time, the entropy decreases, and so no collisions can occur to the past, only in the future, which is of course consistent with the attractive nature of the dynamics. As is a convex function of the positions , one expects to also evolve in a convex manner in time, that is to say the energy should be increasing. This is indeed the case:

Exercise 1Show that

Symmetric polynomials of the zeroes are polynomial functions of the coefficients and should thus evolve in a polynomial fashion. One can compute this explicitly in simple cases. For instance, the center of mass is an invariant:

The variance decreases linearly:

Exercise 2Establish the virial identity

As the variance (which is proportional to ) cannot become negative, this identity shows that “finite time blowup” must occur – that the zeroes must collide at or before the time .

Exercise 3Show that theStieltjes transformsolves the viscous Burgers equation

either by using the original heat equation (2) and the identity , or else by using the equations of motion (3). This relation between the Burgers equation and the heat equation is known as the Cole-Hopf transformation.

The paper of Csordas, Smith, and Varga mentioned previously gives some other bounds on the lifespan of the dynamics; roughly speaking, they show that if there is one pair of zeroes that are much closer to each other than to the other zeroes then they must collide in a short amount of time (unless there is a collision occuring even earlier at some other location). Their argument extends also to situations where there are an infinite number of zeroes, which they apply to get new results on Newman’s conjecture in analytic number theory. I would be curious to know of further places in the literature where this dynamics has been studied.

The equidistribution theorem asserts that if is an irrational phase, then the sequence is equidistributed on the unit circle, or equivalently that

for any continuous (or equivalently, for any smooth) function . By approximating uniformly by a Fourier series, this claim is equivalent to that of showing that

for any non-zero integer (where ), which is easily verified from the irrationality of and the geometric series formula. Conversely, if is rational, then clearly fails to go to zero when is a multiple of the denominator of .

One can then ask for more quantitative information about the decay of exponential sums of , or more generally on exponential sums of the form for an arithmetic progression (in this post all progressions are understood to be finite) and a polynomial . It will be convenient to phrase such information in the form of an *inverse theorem*, describing those phases for which the exponential sum is large. Indeed, we have

Lemma 1 (Geometric series formula, inverse form)Let be an arithmetic progression of length at most for some , and let be a linear polynomial for some . Iffor some , then there exists a subprogression of of size such that varies by at most on (that is to say, lies in a subinterval of of length at most ).

*Proof:* By a linear change of variable we may assume that is of the form for some . We may of course assume that is non-zero in , so that ( denotes the distance from to the nearest integer). From the geometric series formula we see that

and so . Setting for some sufficiently small absolute constant , we obtain the claim.

Thus, in order for a linear phase to fail to be equidistributed on some long progression , must in fact be almost constant on large piece of .

As is well known, this phenomenon generalises to higher order polynomials. To achieve this, we need two elementary additional lemmas. The first relates the exponential sums of to the exponential sums of its “first derivatives” .

Lemma 2 (Van der Corput lemma, inverse form)Let be an arithmetic progression of length at most , and let be an arbitrary function such that

for some . Then, for integers , there exists a subprogression of , of the same spacing as , such that

*Proof:* Squaring (1), we see that

We write and conclude that

where is a subprogression of of the same spacing. Since , we conclude that

for values of (this can be seen, much like the pigeonhole principle, by arguing via contradiction for a suitable choice of implied constants). The claim follows.

The second lemma (which we recycle from this previous blog post) is a variant of the equidistribution theorem.

Lemma 3 (Vinogradov lemma)Let be an interval for some , and let be such that for at least values of , for some . Then eitheror

or else there is a natural number such that

*Proof:* We may assume that and , since we are done otherwise. Then there are at least two with , and by the pigeonhole principle we can find in with and . By the triangle inequality, we conclude that there exists at least one natural number for which

We take to be minimal amongst all such natural numbers, then we see that there exists coprime to and such that

If then we are done, so suppose that . Suppose that are elements of such that and . Writing for some , we have

By hypothesis, ; note that as and we also have . This implies that and thus . We then have

We conclude that for fixed with , there are at most elements of such that . Iterating this with a greedy algorithm, we see that the number of with is at most ; since , this implies that

and the claim follows.

Now we can quickly obtain a higher degree version of Lemma 1:

Proposition 4 (Weyl exponential sum estimate, inverse form)Let be an arithmetic progression of length at most for some , and let be a polynomial of some degree at most . Iffor some , then there exists a subprogression of with such that varies by at most on .

*Proof:* We induct on . The cases are immediate from Lemma 1. Now suppose that , and that the claim had already been proven for . To simplify the notation we allow implied constants to depend on . Let the hypotheses be as in the proposition. Clearly cannot exceed . By shrinking as necessary we may assume that for some sufficiently small constant depending on .

By rescaling we may assume . By Lemma 3, we see that for choices of such that

for some interval . We write , then is a polynomial of degree at most with leading coefficient . We conclude from induction hypothesis that for each such , there exists a natural number such that , by double-counting, this implies that there are integers in the interval such that . Applying Lemma 3, we conclude that either , or that

In the former case the claim is trivial (just take to be a point), so we may assume that we are in the latter case.

We partition into arithmetic progressions of spacing and length comparable to for some large depending on to be chosen later. By hypothesis, we have

so by the pigeonhole principle, we have

for at least one such progression . On this progression, we may use the binomial theorem and (4) to write as a polynomial in of degree at most , plus an error of size . We thus can write for for some polynomial of degree at most . By the triangle inequality, we thus have (for large enough) that

and hence by induction hypothesis we may find a subprogression of of size such that varies by most on , and thus (for large enough again) that varies by at most on , and the claim follows.

This gives the following corollary (also given as Exercise 16 in this previous blog post):

Corollary 5 (Weyl exponential sum estimate, inverse form II)Let be a discrete interval for some , and let polynomial of some degree at most for some . Iffor some , then there is a natural number such that for all .

One can obtain much better exponents here using Vinogradov’s mean value theorem; see Theorem 1.6 this paper of Wooley. (Thanks to Mariusz Mirek for this reference.) However, this weaker result already suffices for many applications, and does not need any result as deep as the mean value theorem.

*Proof:* To simplify notation we allow implied constants to depend on . As before, we may assume that for some small constant depending only on . We may also assume that for some large , as the claim is trivial otherwise (set ).

Applying Proposition 4, we can find a natural number and an arithmetic subprogression of such that and such that varies by at most on . Writing for some interval of length and some , we conclude that the polynomial varies by at most on . Taking order differences, we conclude that the coefficient of this polynomial is ; by the binomial theorem, this implies that differs by at most on from a polynomial of degree at most . Iterating this, we conclude that the coefficient of is for , and the claim then follows by inverting the change of variables (and replacing with a larger quantity such as as necessary).

For future reference we also record a higher degree version of the Vinogradov lemma.

Lemma 6 (Polynomial Vinogradov lemma)Let be a discrete interval for some , and let be a polynomial of degree at most for some such that for at least values of , for some . Then either

or else there is a natural number such that

for all .

*Proof:* We induct on . For this follows from Lemma 3 (noting that if then ), so suppose that and that the claim is already proven for . We now allow all implied constants to depend on .

For each , let denote the number of such that . By hypothesis, , and clearly , so we must have for choices of . For each such , we then have for choices of , so by induction hypothesis, either (5) or (6) holds, or else for choices of , there is a natural number such that

for , where are the coefficients of the degree polynomial . We may of course assume it is the latter which holds. By the pigeonhole principle we may take to be independent of .

Since , we have

for choices of , so by Lemma 3, either (5) or (6) holds, or else (after increasing as necessary) we have

We can again assume it is the latter that holds. This implies that modulo , so that

for choices of . Arguing as before and iterating, we obtain the claim.

The above results also extend to higher dimensions. Here is the higher dimensional version of Proposition 4:

Proposition 7 (Multidimensional Weyl exponential sum estimate, inverse form)Let and , and let be arithmetic progressions of length at most for each . Let be a polynomial of degrees at most in each of the variables separately. Iffor some , then there exists a subprogression of with for each such that varies by at most on .

A much more general statement, in which the polynomial phase is replaced by a nilsequence, and in which one does not necessarily assume the exponential sum is small, is given in Theorem 8.6 of this paper of Ben Green and myself, but it involves far more notation to even state properly.

*Proof:* We induct on . The case was established in Proposition 5, so we assume that and that the claim has already been proven for . To simplify notation we allow all implied constants to depend on . We may assume that for some small depending only on .

By a linear change of variables, we may assume that for all .

We write . First suppose that . Then by the pigeonhole principle we can find such that

and the claim then follows from the induction hypothesis. Thus we may assume that for some large depending only on . Similarly we may assume that for all .

By the triangle inequality, we have

The inner sum is , and the outer sum has terms. Thus, for choices of , one has

for some polynomials of degrees at most in the variables . For each obeying (7), we apply Corollary 5 to conclude that there exists a natural number such that

for (the claim also holds for but we discard it as being trivial). By the pigeonhole principle, there thus exists a natural number such that

for all and for choices of . If we write

where is a polynomial of degrees at most , then for choices of we then have

Applying Lemma 6 in the and the largeness hypotheses on the (and also the assumption that ) we conclude (after enlarging as necessary, and pigeonholing to keep independent of ) that

for all (note that we now include that case, which is no longer trivial) and for choices of . Iterating this, we eventually conclude (after enlarging as necessary) that

whenever for , with nonzero. Permuting the indices, and observing that the claim is trivial for , we in fact obtain (8) for all , at which point the claim easily follows by taking for each .

An inspection of the proof of the above result (or alternatively, by combining the above result again with many applications of Lemma 6) reveals the following general form of Proposition 4, which was posed as Exercise 17 in this previous blog post, but had a slight misprint in it (it did not properly treat the possibility that some of the could be small) and was a bit trickier to prove than anticipated (in fact, the reason for this post was that I was asked to supply a more detailed solution for this exercise):

Proposition 8 (Multidimensional Weyl exponential sum estimate, inverse form, II)Let be an natural number, and for each , let be a discrete interval for some . Letbe a polynomial in variables of multidegrees for some . If

for some , or else there is a natural number such that

Again, the factor of is natural in this bound. In the case, the option (10) may be deleted since (11) trivially holds in this case, but this simplification is no longer available for since one needs (10) to hold for *all* (not just one ) to make (11) completely trivial. Indeed, the above proposition fails for if one removes (10) completely, as can be seen for instance by inspecting the exponential sum , which has size comparable to regardless of how irrational is.

Vitaly Bergelson, Tamar Ziegler, and I have just uploaded to the arXiv our joint paper “Multiple recurrence and convergence results associated to -actions“. This paper is primarily concerned with *limit formulae* in the theory of multiple recurrence in ergodic theory. Perhaps the most basic formula of this type is the *mean ergodic theorem*, which (among other things) asserts that if is a measure-preserving -system (which, in this post, means that is a probability space and is measure-preserving and invertible, thus giving an action of the integers), and are functions, and is ergodic (which means that contains no -invariant functions other than the constants (up to almost everywhere equivalence, of course)), then the average

converges as to the expression

see e.g. this previous blog post. Informally, one can interpret this limit formula as an equidistribution result: if is drawn at random from (using the probability measure ), and is drawn at random from for some large , then the pair becomes uniformly distributed in the product space (using product measure ) in the limit as .

If we allow to be non-ergodic, then we still have a limit formula, but it is a bit more complicated. Let be the -invariant measurable sets in ; the -system can then be viewed as a *factor* of the original system , which is equivalent (in the sense of measure-preserving systems) to a trivial system (known as the *invariant factor*) in which the shift is trivial. There is then a projection map to the invariant factor which is a factor map, and the average (1) converges in the limit to the expression

where is the pushforward map associated to the map ; see e.g. this previous blog post. We can interpret this as an equidistribution result. If is a pair as before, then we no longer expect complete equidistribution in in the non-ergodic, because there are now non-trivial constraints relating with ; indeed, for any -invariant function , we have the constraint ; putting all these constraints together we see that (for almost every , at least). The limit (2) can be viewed as an assertion that this constraint are in some sense the “only” constraints between and , and that the pair is uniformly distributed relative to these constraints.

Limit formulae are known for multiple ergodic averages as well, although the statement becomes more complicated. For instance, consider the expression

for three functions ; this is analogous to the combinatorial task of counting length three progressions in various sets. For simplicity we assume the system to be ergodic. Naively one might expect this limit to then converge to

which would roughly speaking correspond to an assertion that the triplet is asymptotically equidistributed in . However, even in the ergodic case there can be additional constraints on this triplet that cannot be seen at the level of the individual pairs , . The key obstruction here is that of *eigenfunctions* of the shift , that is to say non-trivial functions that obey the eigenfunction equation almost everywhere for some constant (or -invariant) . Each such eigenfunction generates a constraint

tying together , , and . However, it turns out that these are in some sense the *only* constraints on that are relevant for the limit (3). More precisely, if one sets to be the sub-algebra of generated by the eigenfunctions of , then it turns out that the factor is isomorphic to a shift system known as the *Kronecker factor*, for some compact abelian group and some (irrational) shift ; the factor map pushes eigenfunctions forward to (affine) characters on . It is then known that the limit of (3) is

where is the closed subgroup

and is the Haar probability measure on ; see this previous blog post. The equation defining corresponds to the constraint (4) mentioned earlier. Among other things, this limit formula implies *Roth’s theorem*, which in the context of ergodic theory is the assertion that the limit (or at least the limit inferior) of (3) is positive when is non-negative and not identically vanishing.

If one considers a quadruple average

(analogous to counting length four progressions) then the situation becomes more complicated still, even in the ergodic case. In addition to the (linear) eigenfunctions that already showed up in the computation of the triple average (3), a new type of constraint also arises from *quadratic eigenfunctions* , which obey an eigenfunction equation in which is no longer constant, but is now a linear eigenfunction. For such functions, behaves quadratically in , and one can compute the existence of a constraint

between , , , and that is not detected at the triple average level. As it turns out, this is not the only type of constraint relevant for (5); there is a more general class of constraint involving two-step nilsystems which we will not detail here, but see e.g. this previous blog post for more discussion. Nevertheless there is still a similar limit formula to previous examples, involving a special factor which turns out to be an inverse limit of two-step nilsystems; this limit theorem can be extracted from the structural theory in this paper of Host and Kra combined with a limit formula for nilsystems obtained by Lesigne, but will not be reproduced here. The pattern continues to higher averages (and higher step nilsystems); this was first done explicitly by Ziegler, and can also in principle be extracted from the structural theory of Host-Kra combined with nilsystem equidistribution results of Leibman. These sorts of limit formulae can lead to various recurrence results refining Roth’s theorem in various ways; see this paper of Bergelson, Host, and Kra for some examples of this.

The above discussion was concerned with -systems, but one can adapt much of the theory to measure-preserving -systems for other discrete countable abelian groups , in which one now has a family of shifts indexed by rather than a single shift, obeying the compatibility relation . The role of the intervals in this more general setting is replaced by that of Folner sequences. For arbitrary countable abelian , the theory for double averages (1) and triple limits (3) is essentially identical to the -system case. But when one turns to quadruple and higher limits, the situation becomes more complicated (and, for arbitrary , still not fully understood). However one model case which is now well understood is the finite field case when is an infinite-dimensional vector space over a finite field (with the finite subspaces then being a good choice for the Folner sequence). Here, the analogue of the structural theory of Host and Kra was worked out by Vitaly, Tamar, and myself in these previous papers (treating the high characteristic and low characteristic cases respectively). In the finite field setting, it turns out that nilsystems no longer appear, and one only needs to deal with linear, quadratic, and higher order eigenfunctions (known collectively as *phase polynomials*). It is then natural to look for a limit formula that asserts, roughly speaking, that if is drawn at random from a -system and drawn randomly from a large subspace of , then the only constraints between are those that arise from phase polynomials. The main theorem of this paper is to establish this limit formula (which, again, is a little complicated to state explicitly and will not be done here). In particular, we establish for the first time that the limit actually exists (a result which, for -systems, was one of the main results of this paper of Host and Kra).

As a consequence, we can recover finite field analogues of most of the results of Bergelson-Host-Kra, though interestingly some of the counterexamples demonstrating sharpness of their results for -systems (based on Behrend set constructions) do not seem to be present in the finite field setting (cf. this previous blog post on the cap set problem). In particular, we are able to largely settle the question of when one has a Khintchine-type theorem that asserts that for any measurable set in an ergodic -system and any , one has

for a syndetic set of , where are distinct residue classes. It turns out that Khintchine-type theorems always hold for (and for ergodicity is not required), and for it holds whenever form a parallelogram, but not otherwise (though the counterexample here was such a painful computation that we ended up removing it from the paper, and may end up putting it online somewhere instead), and for larger we could show that the Khintchine property failed for generic choices of , though the problem of determining exactly the tuples for which the Khintchine property failed looked to be rather messy and we did not completely settle it.

One of the first non-trivial theorems one encounters in classical algebraic geometry is Bézout’s theorem, which we will phrase as follows:

Theorem 1 (Bézout’s theorem)Let be a field, and let be non-zero polynomials in two variables with no common factor. Then the two curves and have no common components, and intersect in at most points.

This theorem can be proven by a number of means, for instance by using the classical tool of resultants. It has many strengthenings, generalisations, and variants; see for instance this previous blog post on Bézout’s inequality. Bézout’s theorem asserts a fundamental algebraic dichotomy, of importance in combinatorial incidence geometry: any two algebraic curves either share a common component, or else have a bounded finite intersection; there is no intermediate case in which the intersection is unbounded in cardinality, but falls short of a common component. This dichotomy is closely related to the integrality gap in algebraic dimension: an algebraic set can have an integer dimension such as or , but cannot attain any intermediate dimensions such as . This stands in marked contrast to sets of analytic, combinatorial, or probabilistic origin, whose “dimension” is typically not necessarily constrained to be an integer.

Bézout’s inequality tells us, roughly speaking, that the intersection of a curve of degree and a curve of degree forms a set of at most points. One can consider the converse question: given a set of points in the plane , can one find two curves of degrees with and no common components, whose intersection contains these points?

A model example that supports the possibility of such a converse is a grid that is a Cartesian product of two finite subsets of with . In this case, one can take one curve to be the union of vertical lines, and the other curve to be the union of horizontal lines, to obtain the required decomposition. Thus, if the proposed converse to Bézout’s inequality held, it would assert that any set of points was essentially behaving like a “nonlinear grid” of size .

Unfortunately, the naive converse to Bézout’s theorem is false. A counterexample can be given by considering a set of points for some large perfect square , where is a by grid of the form described above, and consists of points on an line (e.g. a or grid). Each of the two component sets can be written as the intersection between two curves whose degrees multiply up to ; in the case of , we can take the two families of parallel lines (viewed as reducible curves of degree ) as the curves, and in the case of , one can take as one curve, and the graph of a degree polynomial on vanishing on for the second curve. But, if is large enough, one cannot cover by the intersection of a single pair of curves with no common components whose degrees multiply up to . Indeed, if this were the case, then without loss of generality we may assume that , so that . By Bézout’s theorem, either contains , or intersects in at most points. Thus, in order for to capture all of , it must contain , which forces to not contain . But has to intersect in points, so by Bézout’s theorem again we have , thus . But then (by more applications of Bézout’s theorem) can only capture of the points of , a contradiction.

But the above counterexample suggests that even if an arbitrary set of (or ) points cannot be covered by the single intersection of a pair of curves with degree multiplying up to , one may be able to cover such a set by a small number of such intersections. The purpose of this post is to record the simple observation that this is, indeed, the case:

Theorem 2 (Partial converse to Bézout’s theorem)Let be a field, and let be a set of points in for some . Then one can find and pairs of coprime non-zero polynomials for such that

Informally, every finite set in the plane is (a dense subset of) the union of logarithmically many nonlinear grids. The presence of the logarithm is necessary, as can be seen by modifying the example to be the union of logarithmically many Cartesian products of distinct dimensions, rather than just a pair of such products.

Unfortunately I do not know of any application of this converse, but I thought it was cute anyways. The proof is given below the fold.

One of the basic problems in analytic number theory is to obtain bounds and asymptotics for sums of the form

in the limit , where ranges over natural numbers less than , and is some arithmetic function of number-theoretic interest. (It is also often convenient to replace this sharply truncated sum with a smoother sum such as , but we will not discuss this technicality here.) For instance, the prime number theorem is equivalent to the assertion

where is the von Mangoldt function, while the Riemann hypothesis is equivalent to the stronger assertion

It is thus of interest to develop techniques to estimate such sums . Of course, the difficulty of this task depends on how “nice” the function is. The functions that come up in number theory lie on a broad spectrum of “niceness”, with some particularly nice functions being quite easy to sum, and some being insanely difficult.

At the easiest end of the spectrum are those functions that exhibit some sort of regularity or “smoothness”. Examples of smoothness include “Archimedean” smoothness, in which is the restriction of some smooth function from the reals to the natural numbers, and the derivatives of are well controlled. A typical example is

One can already get quite good bounds on this quantity by comparison with the integral , namely

with sharper bounds available by using tools such as the Euler-Maclaurin formula (see this blog post). Exponentiating such asymptotics, incidentally, leads to one of the standard proofs of Stirling’s formula (as discussed in this blog post).

One can also consider “non-Archimedean” notions of smoothness, such as periodicity relative to a small period . Indeed, if is periodic with period (and is thus essentially a function on the cyclic group ), then one has the easy bound

In particular, we have the fundamental estimate

This is a good estimate when is much smaller than , but as approaches in magnitude, the error term begins to overwhelm the main term , and one needs much more delicate information on the fractional part of in order to obtain good estimates at this point.

One can also consider functions which combine “Archimedean” and “non-Archimedean” smoothness into an “adelic” smoothness. We will not define this term precisely here (though the concept of a Schwartz-Bruhat function is one way to capture this sort of concept), but a typical example might be

where is periodic with some small period . By using techniques such as summation by parts, one can estimate such sums using the techniques used to estimate sums of periodic functions or functions with (Archimedean) smoothness.

Another class of functions that is reasonably well controlled are the multiplicative functions, in which whenever are coprime. Here, one can use the powerful techniques of multiplicative number theory, for instance by working with the Dirichlet series

which are clearly related to the partial sums (essentially via the Mellin transform, a cousin of the Fourier and Laplace transforms); for this post we ignore the (important) issue of how to make sense of this series when it is not absolutely convergent (but see this previous blog post for more discussion). A primary reason that this technique is effective is that the Dirichlet series of a multiplicative function factorises as an Euler product

One also obtains similar types of representations for functions that are not quite multiplicative, but are closely related to multiplicative functions, such as the von Mangoldt function (whose Dirichlet series is not given by an Euler product, but instead by the logarithmic derivative of an Euler product).

Moving another notch along the spectrum between well-controlled and ill-controlled functions, one can consider functions that are *divisor sums* such as

for some other arithmetic function , and some *level* . This is a linear combination of periodic functions and is thus *technically* periodic in (with period equal to the least common multiple of all the numbers from to ), but in practice this periodic is far too large to be useful (except for extremely small levels , e.g. ). Nevertheless, we can still control the sum simply by rearranging the summation:

and thus by (1) one can bound this by the sum of a main term and an error term . As long as the level is significantly less than , one may expect the main term to dominate, and one can often estimate this term by a variety of techniques (for instance, if is multiplicative, then multiplicative number theory techniques are quite effective, as mentioned previously). Similarly for other slight variants of divisor sums, such as expressions of the form

or expressions of the form

where each is periodic with period .

One of the simplest examples of this comes when estimating the divisor function

which counts the number of divisors up to . This is a multiplicative function, and is therefore most efficiently estimated using the techniques of multiplicative number theory; but for reasons that will become clearer later, let us “forget” the multiplicative structure and estimate the above sum by more elementary methods. By applying the preceding method, we see that

Here, we are (barely) able to keep the error term smaller than the main term; this is right at the edge of the divisor sum method, because the level in this case is equal to . Unfortunately, at this high choice of level, it is not always possible to always keep the error term under control like this. For instance, if one wishes to use the standard divisor sum representation

where is the Möbius function, to compute , then one ends up looking at

From Dirichlet series methods, it is not difficult to establish the identities

and

This suggests (but does not quite prove) that one has

in the sense of conditionally convergent series. Assuming one can justify this (which, ultimately, requires one to exclude zeroes of the Riemann zeta function on the line , as discussed in this previous post), one is eventually left with the estimate , which is useless as a lower bound (and recovers only the classical Chebyshev estimate as the upper bound). The inefficiency here when compared to the situation with the divisor function can be attributed to the signed nature of the Möbius function , which causes some cancellation in the divisor sum expansion that needs to be compensated for with improved estimates.

However, there are a number of tricks available to reduce the level of divisor sums. The simplest comes from exploiting the change of variables , which can in principle reduce the level by a square root. For instance, when computing the divisor function , one can observe using this change of variables that every divisor of above is paired with one below , and so we have

except when is a perfect square, in which case one must subtract one from the right-hand side. Using this reduced-level divisor sum representation, one can obtain an improvement to (2), namely

This type of argument is also known as the Dirichlet hyperbola method. A variant of this argument can also deduce the prime number theorem from (3), (4) (and with some additional effort, one can even drop the use of (4)); this is discussed at this previous blog post.

Using this square root trick, one can now also control divisor sums such as

(Note that has no multiplicativity properties in , and so multiplicative number theory techniques cannot be directly applied here.) The level of the divisor sum here is initially of order , which is too large to be useful; but using the square root trick, we can expand this expression as

which one can rewrite as

The constraint is periodic in with period , so we can write this as

where is the number of solutions in to the equation , and so

The function is multiplicative, and can be easily computed at primes and prime powers using tools such as quadratic reciprocity and Hensel’s lemma. For instance, by Fermat’s two-square theorem, is equal to for and for . From this and standard multiplicative number theory methods (e.g. by obtaining asymptotics on the Dirichlet series ), one eventually obtains the asymptotic

and also

and thus

Similar arguments give asymptotics for on other quadratic polynomials; see for instance this paper of Hooley and these papers by McKee. Note that the irreducibility of the polynomial will be important. If one considers instead a sum involving a reducible polynomial, such as , then the analogous quantity becomes significantly larger, leading to a larger growth rate (of order rather than ) for the sum.

However, the square root trick is insufficient by itself to deal with higher order sums involving the divisor function, such as

the level here is initially of order , and the square root trick only lowers this to about , creating an error term that overwhelms the main term. And indeed, the asymptotic for such this sum has not yet been rigorously established (although if one heuristically drops error terms, one can arrive at a reasonable conjecture for this asymptotic), although some results are known if one averages over additional parameters (see e.g. this paper of Greaves, or this paper of Matthiesen).

Nevertheless, there is an ingenious argument of Erdös that allows one to obtain good *upper* and *lower* bounds for these sorts of sums, in particular establishing the asymptotic

for any *fixed* irreducible non-constant polynomial that maps to (with the implied constants depending of course on the choice of ). There is also the related moment bound

for any fixed (not necessarily irreducible) and any fixed , due to van der Corput; this bound is in fact used to dispose of some error terms in the proof of (6). These should be compared with what one can obtain from the divisor bound and the trivial bound , giving the bounds

for any fixed .

The lower bound in (6) is easy, since one can simply lower the level in (5) to obtain the lower bound

for any , and the preceding methods then easily allow one to obtain the lower bound by taking small enough (more precisely, if has degree , one should take equal to or less). The upper bounds in (6) and (7) are more difficult. Ideally, if we could obtain upper bounds of the form

for any fixed , then the preceding methods would easily establish both results. Unfortunately, this bound can fail, as illustrated by the following example. Suppose that is the product of distinct primes , each of which is close to . Then has divisors, with of them close to for each . One can think of (the logarithms of) these divisors as being distributed according to what is essentially a Bernoulli distribution, thus a randomly selected divisor of has magnitude about , where is a random variable which has the same distribution as the number of heads in independently tossed fair coins. By the law of large numbers, should concentrate near when is large, which implies that the majority of the divisors of will be close to . Sending , one can show that the bound (8) fails whenever .

This however can be fixed in a number of ways. First of all, even when , one can show weaker substitutes for (8). For instance, for any fixed and one can show a bound of the form

for some depending only on . This nice elementary inequality (first observed by Landreau) already gives a quite short proof of van der Corput’s bound (7).

For Erdös’s upper bound (6), though, one cannot afford to lose these additional factors of , and one must argue more carefully. Here, the key observation is that the counterexample discussed earlier – when the natural number is the product of a large number of fairly small primes – is quite atypical; most numbers have at least one large prime factor. For instance, the number of natural numbers less than that contain a prime factor between and is equal to

which, thanks to Mertens’ theorem

for some absolute constant , is comparable to . In a similar spirit, one can show by similarly elementary means that the number of natural numbers less than that are *-smooth*, in the sense that all prime factors are at most , is only about or so. Because of this, one can hope that the bound (8), while not true in full generality, will still be true for *most* natural numbers , with some slightly weaker substitute available (such as (7)) for the exceptional numbers . This turns out to be the case by an elementary but careful argument.

The Erdös argument is quite robust; for instance, the more general inequality

for fixed irreducible and , which improves van der Corput’s inequality (8) was shown by Delmer using the same methods. (A slight error in the original paper of Erdös was also corrected in this latter paper.) In a forthcoming revision to my paper on the Erdös-Straus conjecture, Christian Elsholtz and I have also applied this method to obtain bounds such as

which turn out to be enough to obtain the right asymptotics for the number of solutions to the equation .

Below the fold I will provide some more details of the arguments of Landreau and of Erdös.

I have blogged several times in the past about nonstandard analysis, which among other things is useful in allowing one to import tools from infinitary (or qualitative) mathematics in order to establish results in finitary (or quantitative) mathematics. One drawback, though, to using nonstandard analysis methods is that the bounds one obtains by such methods are usually *ineffective*: in particular, the conclusions of a nonstandard analysis argument may involve an unspecified constant that is known to be finite but for which no explicit bound is obviously available. (In many cases, a bound can eventually be worked out by performing *proof mining* on the argument, and in particular by carefully unpacking the *proofs* of all the various results from infinitary mathematics that were used in the argument, as opposed to simply using them as “black boxes”, but this is a time-consuming task and the bounds that one eventually obtains tend to be quite poor (e.g. tower exponential or Ackermann type bounds are not uncommon).)

Because of this fact, it would seem that quantitative bounds, such as polynomial type bounds that show that one quantity is controlled in a polynomial fashion by another quantity , are not easily obtainable through the ineffective methods of nonstandard analysis. Actually, this is not the case; as I will demonstrate by an example below, nonstandard analysis can certainly yield polynomial type bounds. The catch is that the exponent in such bounds will be ineffective; but nevertheless such bounds are still good enough for many applications.

Let us now illustrate this by reproving a lemma from this paper of Mei-Chu Chang (Lemma 2.14, to be precise), which was recently pointed out to me by Van Vu. Chang’s paper is focused primarily on the sum-product problem, but she uses a quantitative lemma from algebraic geometry which is of independent interest. To motivate the lemma, let us first establish a qualitative version:

Lemma 1 (Qualitative solvability)Let be a finite number of polynomials in several variables with rational coefficients. If there is a complex solution to the simultaneous system of equationsthen there also exists a solution whose coefficients are algebraic numbers (i.e. they lie in the algebraic closure of the rationals).

*Proof:* Suppose there was no solution to over . Applying Hilbert’s nullstellensatz (which is available as is algebraically closed), we conclude the existence of some polynomials (with coefficients in ) such that

as polynomials. In particular, we have

for all . This shows that there is no solution to over , as required.

Remark 1Observe that in the above argument, one could replace and by any other pair of fields, with the latter containing the algebraic closure of the former, and still obtain the same result.

The above lemma asserts that if a system of rational equations is solvable at all, then it is solvable with some algebraic solution. But it gives no bound on the complexity of that solution in terms of the complexity of the original equation. Chang’s lemma provides such a bound. If is an integer, let us say that an algebraic number has height at most if its minimal polynomial (after clearing denominators) consists of integers of magnitude at most .

Lemma 2 (Quantitative solvability)Let be a finite number of polynomials of degree at most with rational coefficients, each of height at most . If there is a complex solution to the simultaneous system of equationsthen there also exists a solution whose coefficients are algebraic numbers of degree at most and height at most , where depends only on , and .

Chang proves this lemma by essentially establishing a quantitative version of the nullstellensatz, via elementary elimination theory (somewhat similar, actually, to the approach I took to the nullstellensatz in my own blog post). She also notes that one could also establish the result through the machinery of Gröbner bases. In each of these arguments, it was not possible to use Lemma 1 (or the closely related nullstellensatz) as a black box; one actually had to unpack one of the proofs of that lemma or nullstellensatz to get the polynomial bound. However, using nonstandard analysis, it is possible to get such polynomial bounds (albeit with an ineffective value of the constant ) directly from Lemma 1 (or more precisely, the generalisation in Remark 1) *without* having to inspect the proof, and instead simply using it as a black box, thus providing a “soft” proof of Lemma 2 that is an alternative to the “hard” proofs mentioned above.

Here’s how the proof works. Informally, the idea is that Lemma 2 should follow from Lemma 1 after replacing the field of rationals with “the field of rationals of polynomially bounded height”. Unfortunately, the latter object does not really make sense as a field in standard analysis; nevertheless, it is a perfectly sensible object in nonstandard analysis, and this allows the above informal argument to be made rigorous.

We turn to the details. As is common whenever one uses nonstandard analysis to prove finitary results, we use a “compactness and contradiction” argument (or more precisely, an “ultralimit and contradiction” argument). Suppose for contradiction that Lemma 2 failed. Carefully negating the quantifiers (and using the axiom of choice), we conclude that there exists such that for each natural number , there is a positive integer and a family of polynomials of degree at most and rational coefficients of height at most , such that there exist at least one complex solution to

but such that there does not exist any such solution whose coefficients are algebraic numbers of degree at most and height at most .

Now we take ultralimits (see e.g. this previous blog post of a quick review of ultralimit analysis, which we will assume knowledge of in the argument that follows). Let be a non-principal ultrafilter. For each , the ultralimit

of the (standard) polynomials is a nonstandard polynomial of degree at most , whose coefficients now lie in the nonstandard rationals . Actually, due to the height restriction, we can say more. Let be the ultralimit of the , this is a nonstandard natural number (which will almost certainly be unbounded, but we will not need to use this). Let us say that a nonstandard integer is *of polynomial size* if we have for some standard natural number , and say that a nonstandard rational number is *of polynomial height* if , are of polynomial size. Let be the collection of all nonstandard rationals of polynomial height. (In the language of nonstandard analysis, is an *external* set rather than an internal one, because it is not itself an ultraproduct of standard sets; but this will not be relevant for the argument that follows.) It is easy to see that is a field, basically because the sum or product of two integers of polynomial size, remains of polynomial size. By construction, it is clear that the coefficients of are nonstandard rationals of polynomial height, and thus are defined over .

Meanwhile, if we let be the ultralimit of the solutions in (1), we have

thus are solvable in . Applying Lemma 1 (or more precisely, the generalisation in Remark 1), we see that are also solvable in . (Note that as is algebraically closed, is also (by Los’s theorem), and so contains .) Thus, there exists with

As lies in , we can write as an ultralimit of standard complex vectors . By construction, the coefficients of each obey a non-trivial polynomial equation of degree at most and whose coefficients are nonstandard integers of magnitude at most , for some standard natural number . Undoing the ultralimit, we conclude that for sufficiently close to , the coefficients of obey a non-trivial polynomial equation of degree at most whose coefficients are *standard* integers of magnitude at most . In particular, these coefficients have height at most . Also, we have

But for larger than , this contradicts the construction of the , and the claim follows. (Note that as is non-principal, any neighbourhood of in will contain arbitrarily large natural numbers.)

Remark 2The same argument actually gives a slightly stronger version of Lemma 2, namely that the integer coefficients used to define the algebraic solution can be taken to be polynomials in the coefficients of , with degree and coefficients bounded by .

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