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ASYMPTOTICS OF THE COEFFICIENTS OF BIVARIATE ANALYTIC FUNCTIONS WITHALGEBRAIC SINGULARITIES (PREPRINT)

TORIN GREENWOOD

ABSTRACT. Flajolet and Odlyzko (1990) derived asymptotic formulae the coefficients of aclass of univariate generating functions with algebraic singularities. These results havebeen extended to classes of multivariate generating functions by Gao and Richmond (1992)and Hwang (1996, 1998), in both cases by reducing the multivariate case to the univariatecase. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and usedthem to analyze the coefficients of rational generating functions. In this paper, we usethese multivariate analytic techniques to find asymptotic formulae for the coefficients of abroad class of bivariate generating functions with algebraic singularities.

1. INTRODUCTION

1.1. Historical Background. Generating functions are a powerful, convenient tool to en-code an array of numbers into a single function. Given that a generating function can becomputed with limited information about the corresponding array, it is often desirable tolearn more about the array from the generating function itself. One particularly useful goalis to approximate the coefficients of a generating function asymptotically as their indicesgrow in a prescribed way.

Work on these asymptotic approximations began with univariate generating functions,which encode sequences of numbers. In 1990, Flajolet and Odlyzko, [3], described howto compute the asymptotics of a class of univariate generating functions with algebraicsingularities. In particular, they considered functions of the form,

(1) g (z) = K (1− z)α(log(1− z)

)γ (loglog(1− z)

)δ ,

where α,γ,δ, and K are arbitrary real numbers. They also considered related classes offunctions, which were again the products and compositions of power functions and loga-rithms. Their results differed from previous results both in the class of generating func-tions covered, and in their method of proof. To show their results, they relied on theCauchy integral formula: [

zn]g (z) = 1

2πi

ˆC

g (z)d z

zn+1

Here, [zn] g (z) represents the coefficient of zn in the power series expansion of g , andC is any positively-oriented contour around the origin which does not enclose any sin-gularities of g (z). Starting with any function f such that f (z) = O (|1− z|α) as z → 1 and

Date: December 19, 2014.2010 Mathematics Subject Classification. 05A15, 05A16.Key words and phrases. Multivariate generating functions, coefficients, asymptotics, algebraic singulari-

ties, smooth critical points, Cauchy integral formula.Submitted to OJAC in 2014. See analytic-combinatorics.org.This article is licensed under a Creative Commons Attribution 4.0 International license.

1

2 TORIN GREENWOOD

picking an explicit contour for C , Flajolet and Odlyzko were able to analyze the Cauchy in-tegral by breaking the domain of integration into several segments. They concluded that[zn] f (z) =O

(n−α−1

). Then, they expanded their results to functions g (z) with the form in

(1).Later in the 1990s, other researchers extended these results to classes of multivariate

generating functions. Bender and Richmond, [2], had already considered the asymptoticsof multivariate generating functions with poles in 1983. In 1992, Gao and Richmond, [4],considered classes of bivariate generating functions F (z, s) which are of a form they calledalgebraico-logrithmic, which includes some generating functions with algebraic singulari-ties. The convenience of algebraico-logrithmic functions is that by considering [zn]F (z, x)and temporarily fixing x, the problem is reduced to a univariate generating function wherethe results of Flajolet and Odlyzko can be applied. Then, the asymptotic approximationsfor [zn]F (z, x) can be broken down further to approximate the coefficients

[zn xk

]F (z, x).

In his 1996 and 1998 papers, [6] and [7], Hwang expanded upon the multivariate results,using a probability framework and deriving large deviation theorems. In 1996, Hwangconsidered sequences of random variables Ωnn . Assuming that the moment generatingfunctions of the Ωn were of a particular form, Hwang proved a central limit theorem forΩnn . Then, he considered a class of bivariate generating functions P (w, z) such that afterapproximating [zn]P (w, z) with Flajolet and Odlyzko’s univariate results, [zn]P (w, z) sat-isfied the same conditions he required previously of the moment generating functions ofΩn . Applying his central limit theorem from before gave asymptotic results for a new classof bivariate generating functions. In 1998, Hwang extended his results by using univariatesaddle-point methods to approximate integrals.

1.2. New Techniques. In their 2013 book, [10], Pemantle and Wilson outline a programwhich greatly extends the results of previous work on multivariate generating functionanalysis. Instead of reducing multivariate generating functions to the one-dimensionalcase, they start with the multivariate version of Cauchy’s integral formula. The domainof integration can be shifted within the region where the generating function is analytic.According to Morse theory, as the domain of integration is shifted, its topology will notchange until it encounters a critical point of a height function h, defined below in sec-tion 2. The integrand in the integral formula decays exponentially as the domain of inte-gration expands beyond critical points, causing the main contribution to the integral tocome from integrating over quasi-local cycles near these critical points. Morse theory alsodescribes the topology of the domain of integration when it is pushed near these criticalpoints, leaving precise integrals that give the dominating asymptotic contributions to thecoefficients of the generating function.

These methods have already been used to analyze many rational generating functions.In their 2002 paper, [8], Pemantle and Wilson analyzed generating functions with smoothcritical points by describing the quasi-local cycles in this situation and computing theresidues of the resulting integrals. In their 2004 paper, [9], they extend their results tothe case where the critical points are multiple points, which changes the topology of thequasi-local cycle. Then, in [1] from 2011, Baryshnikov and Pemantle examined the case ofquadratic points.

This paper concerns generating functions with algebraic singularities and smooth crit-ical points. We will examine functions of the form, H(x, y)−β, where H is an analytic func-tion andβ is an arbitrary real number. Using the procedure outlined in [10], one sees again

COEFFICIENTS OF BIVARIATE ANALYTIC FUNCTIONS WITH ALGEBRAIC SINGULARITIES 3

that the coefficients of H−β are well-approximated by the sum of integrals over quasi-localcycles around the critical points of the function H . However, instead of computing theresidues of the remaining integrals, we use classical contour deformations and analyzethe integrand directly. Hankel contours historically enabled the analysis of integrals indomains with branch cuts. Here, we will use a modified version of such a contour, andwe will show that H−β is well-approximated by a one-dimensional binomial function overpart of this contour. This one-dimensional function is easy to integrate, giving us the mainresult stated in section 3 below.

2. MULTIVARIATE ANALYTIC COMBINATORICS

Consider any analytic function H(z), z ∈ Cd , and any β ∈ R. We will approximate thecoefficients [zr] H(z)−β as r approaches infinity along some ray inZd . In order to do so, wewill use the multivariate version of Cauchy’s integral formula:

(2)[zr]H(z)−β =

(1

2πi

)2 ÏT

H(z)−βz−r−1 dx dy.

Here, the torus T encloses zero, but does not enclose any singularities of the functionH(z)−β. In order to approximate this integral effectively, we would like to identify wherethe integrand is large. We choose a simple height function,

h(z) := hr(z) :=−r ·Re logz.

Here, r is the unit vector in the direction of r. h will approximate the log magnitude of theintegrand in the Cauchy integral. Although it excludes the contribution from z−1H(z), thisterm is bounded on compact sets and thus can be omitted as we consider large r.

Now, we restrict our attention of this height function momentarily to V := VH , the zeroset of H . As described in Goresky and MacPherson’s book on stratified Morse theory, [5],if V has a Whitney stratification, then the only places where the homology group of V

can change are the critical points of h. This means that when taking an integral over acontour in V , we can move the contour freely until it reaches the critical points, becausethe homology of the cycle does not change unless it hits a critical point. Note that theintegral of the same integrand over two homologous cycles is not necessarily zero, unlikefor homotopic cycles. However, these integrals can be shown to differ by an exponentiallysmall error term. Thus, our goal will be to expand the domain of integration to minimizethe maximum values of h. The largest values of h will occur near a set of critical points.

So far, our discussion referred to cycles within V itself, but our integral is over the torusin M =Cd \V . However, there is a dual version of the Morse theoretic statements discussedabove which states that the homology classes of M only change at these same criticalpoints of h.

Now, we must find the critical points of h. First, we consider a stratification of the spaceVH , and restrict our attention to critical points within a certain stratum S. Given S, we maywrite the closure S as the intersection of several other varieties V f1 , . . . ,V fd−k , where the f j

are computable and have non-vanishing gradients at every point of S. Note that while S isthe actual stratum, S is a k-dimensional variety containing the stratum that may be largerthan S.

Let z = (z1, . . . , zd ). To find the critical points, we look for where dh|S vanishes. dh|Svanishing at z is equivalent to the vector r being in the span of the d −k vectors ∇log fi (z) :1 ≤ i ≤ d −k, where ∇log f (z) := (z1∂ f /∂z1, . . . , zd∂ f /∂zd ) is the gradient of f with respect

4 TORIN GREENWOOD

to log z. If we let M be the (d − k + 1)× d matrix whose rows are these d − k gradientsand r, then at all points of S the submatrix of M consisting of the first d −k rows has rankd −k. The span of the d −k gradients containing r is equivalent to the vanishing of the kdeterminants Md−k+i , where Md−k+i contains the first d −k columns of M together withthe (d −k + i )th column. This gives the d critical point equations:

fi = 0, det(Md−k+i ) = 0

Smooth critical points are a simple yet common special case, where the stratum S isof dimension k := d − 1, and S = V . Thus, in this case, f1 = H = 0, and the other d − 1equations from above can be rewritten as follows:

r1z2∂H

∂z2= r2z1

∂H

∂z1

...

r1zd∂H

∂zd= rd z1

∂H

∂z1

This may be written as H = 0,∇logH ||r∗. To check for smoothness at these critical points,we also require that ∇H does not vanish on V . When H is a polynomial, the above criticalpoint equations form a system of polynomial equations. In this case, Gröbner Bases canhelp compute the critical points. In general, it is not necessarily true that all critical pointswill contribute to the leading term of the Cauchy integral.

Goresky and MacPherson show in [5] how Morse theory can lead to an explicit descrip-tion of the domain of integration near a critical point. In the case of generating functionswithout algebraic singularities, this machinery can be used to evaluate the residues of theintegrals near each critical point. However, in the case where H−β has algebraic singular-ities, we will rely on specific homotopies of the contour, and hence we do not need to useMorse theory to determine the domain of integration.

3. NEW RESULT

Let us summarize notation in a bivariate setting. Let V be the zero set of the analyticfunction, H(x, y), where H(0,0) 6= 0. We will approximate the coefficients

[xr y s

]H(x, y)−β

for a fixed β ∈ R as r and s approach infinity with their ratio approaching the constant, λ.Critical points in the direction of λ= r+O(1)

s (as r and s approach infinity) are defined to bethe points (pi , qi ) where the following equations hold:

H = 0

r y∂H

∂y= sx

∂H

∂x

We call the critical points smooth if ∇H does not vanish on V at the critical points. Let D

be the domain of convergence of the power series of H−β that converges around the origin,(0,0). Then, a critical point (p, q) is called minimal if (p, q) ∈ ∂D. We will apply the proce-dure from Section 2 to prove the following result about bivariate generating functions withalgebraic singularities:

Theorem. Let H be an analytic function with exactly n minimal critical points (pi , qi )ni=1,

all of which are smooth and lie on the same torus T ∗. (Hence, |pi | = |p j | and |qi | = |q j |for all 1 ≤ i , j ≤ n.) Assume that these critical points are the only zeroes of H on ∂D, the


boundary of convergence of the power series for H−β. Let λ = r+O(1)s as r, s → ∞. Define

χ1,χ2, and ci as follows:

χ1 = Hy (pi , qi )

Hx(pi , qi )= p

λq

χ2 = 1

2Hx(χ2

1Hxx −2χ1Hx y +Hy y )∣∣∣(x,y)=(pi ,qi )

ci = 1√−2χ2

pi− χ2

1

p2i− 1λq2

i

Assume that the argument of ci does not differ from the argument of qi by a multiple of π,and choose the root in the definition of ci so that qi + tci increases in argument as t > 0increases.

For all i , assume pi , qi , Hx(pi , qi ), and 2χ2pi

+ χ21

pi+ 1

λqiare nonzero. Define

x−β

P as the

principle value of x−β, defined by using a ray from the origin of C as the branch cut of thelogarithm. In this definition, choose any ray such that

H(x, y)−β

P = H(x, y)−β in a neigh-

borhood of the origin in C2 (as defined by the power series of H−β), and such that this raydoes not pass through−pi Hx(pi , qi ) for any i . Letωi be the signed number of times the curveH(t pi , t qi ) crosses this branch cut in a counterclockwise direction as t increases, 0 ≤ t < 1.Then, the following expression holds as r, s →∞ with λ= r+O(1)

s :

[xr y s]H(x, y)−β =−n∑

i=1

i ci rβ−32(−Hx(pi , qi )pi )−β

P e−β(2πiωi )p−r

i q−s−1i

Γ(β)p

2π+o

(rβ−

32 p−r

1 q−s1

)To prove this result, we analyze the multivariate Cauchy integral formula, (2). The dom-

inating contributions to this integral come over quasi-local cycles centered at each criticalpoint, since the height function h decays exponentially away from these points. Becausethe critical points are discrete, we can analyze them individually. An outline of the anal-ysis is as follows: we will show that H(x, y) behaves essentially as a linear function in onevariable, with some minor error terms in the second variable. In order to do this, we willneed to introduce a change of variables, (u, v), which will give us a particularly nice powerseries expansion of H in u and v . This change of variables is what determines χ1 and χ2.Next, we will justify approximating the resulting integral by an iterated integral, relying onthe fact that H is nearly linear. This step is by far the most tedious, and will take manylemmas to justify. Finally, we will analyze this iterated integral.

Note that it is possible to use Morse theory to show that any zeroes of H on ∂D mustbe critical points. Additionally, when all the minimal critical points are smooth, they mustcontribute to the leading term asymptotics of the coefficients of H , which forces each pair(pi , qi ) to contribute a term of equal order in r and s to the asymptotics in the theorem.

4. PROOF SET-UP

4.1. A Convenient Change of Variables. In order to approximate H(x, y) as a univariatelinear function near the critical point (p, q), we will need the power series expansion ofH to have no constant term, linear term, nor quadratic term in one of its two input vari-ables. (We will prove that this condition allows us to approximate H as a univariate linear

6 TORIN GREENWOOD

function later.) With this goal in mind, we define the following change of variables:

u = x +χ1(y −q)+χ2(y −q)2

v = y

Here, χ1 and χ2 are as defined in Theorem 3 above. Write H as a power series in u and v :

H(x, y) = ∑m,n≥0

dmn(u −p)m(v −q)n =: H(u, v)

Since H(p, q) = 0, we have that d00 = 0. Notice that when (x, y) = (p, q), we also have that(u, v) = (p, q). We can check that d01 = d02 = 0:

∂H

∂v

∣∣∣∣(u,v)=(p,q)

=(∂H

∂x· ∂x

∂v+ ∂H

∂y· ∂y

∂v

)∣∣∣∣(x,y)=(p,q)

= Hx · (−χ1 −2χ2(v −q))+Hy∣∣(u,v)=(p,q)

= −χ1Hx +Hy∣∣(u,v)=(p,q)

= 0

∂2H

∂v2

∣∣∣∣(u,v)=(p,q)

= −2χ2Hx + (−χ1 −2χ2(v −q))

[Hxx

∂x

∂v+Hx y

∂y

∂v

]+Hx y

∂x

∂v+Hy y

∂y

∂v

∣∣∣∣(u,v)=(p,q)

= −2χ2Hx +χ21Hxx −2χ1Hx y +Hy y

∣∣(u,v)=(p,q)

= 0

Thus, H(u, v) =∑m,n≥0 dmn(u −p)m(v −q)n with d00 = d01 = d02 = 0.

4.2. Determining the Quasi-Local Cycle. Recall that the original domain of integrationin (2) is a torus T around the origin which encloses no singularities of H−β(x, y). To de-crease the magnitude of the integrand exponentially as r and s approach infinity, we willexpand the torus T towards the minimal critical point, (p, q). Because (p, q) is a minimalcritical point, there cannot be any zeroes between the origin and (p, q) that would other-wise obstruct the deformation. Hence, we can expand the domain of integration througha homotopy until it is near each critical point. Previous work on generating functions withpoles relied on computing residues, but the branch point created by algebraic singular-ities forces us to use explicit contour deformations through homotopies here. In orderto expand the domain of integration past the critical points of a generating function withalgebraic singularities, we must lift to the universal cover of the generating function. How-ever, we will provide explicit contour deformations, as in Figures 1 and 2 below, so it is notnecessary to understand the general method here.

Before expanding the torus, T is the product of a small x circle and a small y circle. Tobegin deforming the torus, shrink the x circle in T so that the y circle may be expandedslightly past q . Then, expand the y circle and morph it so that near q , it is a straight linesegment from the point q − cεy to the point q + cεy for some εy > 0 small enough that thecontour hits no zeroes of H . (c is defined in the Theorem. We will restrict εy further later.)This line segment is the y portion of the quasi-local contour, and is shown in Figure 1.


|q|

q

q - cε

q + cεy

y

Re y

Im y

FIGURE 1. The y portion of the quasi-local contour

|p| + εx

Re x

Im x

p + G(y)

p + G(y)

1/r

γ

γ

γ

γ

γ

1

2

3

4

5

FIGURE 2. The x contour on the left, and a close up of the contour on the right.

Note that the line could be pointed in any direction, depending on the value of c. Forexample, there are instances of critical points for which this line segment is pointing di-rectly towards the origin. We have omitted this case from the statement of the theorem,but it can be handled by perturbing the value of c slightly to determine the appropriateorientation of the line segment.

Now, for each y ∈ [q − cεy , q + cεy ], we will expand the x circle until it approaches thezero set of H near p. When y is close to q , we will wrap the x contour around the zero setof H . However, when y is further away from q , we will expand the x contour less, so that itdoes not come into contact with the zero set of H .

More explicitly, since Hx(p, q) 6= 0 and H is analytic, the implicit function theorem guar-antees that we can parameterize the variety V = (x, y)|H(x, y) = 0 by a smooth functionG(y), so that H(p +G(y), y) = 0 for all y ∈ [q −cεy , q +cεy ] with εy sufficiently small. So, for

|v −q| ≤ εy

2 , we choose the x contour in Figure 2.

8 TORIN GREENWOOD

The equations for the pieces of the contour are as follows:

γ1(y) :=

x : |x −p −G(y)| = 1

r,arg(p) ≤ arg(x −p −G(y)) ≤ arg(p)+2π

γ2(y) :=

x :

1

r≤ |x −p −G(y)| ≤ εx ,arg(x −p −G(y)) = arg(p)+2π

γ3(y) :=

x :

1

r≤ |x −p −G(y)| ≤ εx ,arg(x −p −G(y)) = arg(p)

γ4(y) :=

x : |x −G(y)| = |p|+εx ,arg(p)−θx ≤ arg(x −G(y)) ≤ arg(p)

γ5(y) := x : |x −G(y)| = |p|+εx ,arg(p) ≤ arg(x −G(y)) ≤ arg(p)+θx

Here, εx and θx > 0 are positive real numbers that are small enough that the contour

hits no other zeroes of H and so that G(y) is a valid parameterization of the zero set of Haround this contour. (We will add more restrictions to both later.) In the contour above,the inner and outer rays γ2 and γ3 are directly on top of each other, but they have differingarguments. These segments should be viewed as different segments in the universal coverof H−β. However, the magnitude of the integrand will be the same on γ2 and γ3, and hencethe difference in argument has no impact in the order-of-magnitude computations below.Thus, we will not talk about the impact of the differing arguments until the end of theproof, in Section 6 below. Before this discussion, we treat the argument as a value modulo2π.

Consider the case where |y − q | ≥ 12εy . As |y − q | increases, we must extricate the x

contours from V . To do this, we will gradually shrink the outer radius of the x contours –that is, the radius |p|+ εx in γ4(y) and γ5(y) will shrink until the contour no longer wrapsaround V . Let δ> 0 be a small positive number (which will also be restricted later). We willlinearly interpolate the radius |x−G(y)| in γ4 and γ5 from |p|+εx to |p|−δ. For |y−q | ≥ 1

2εy ,we can write y = q + tc with t ∈ [−1,−1

2

]∪ [12 ,1

]. With this in mind, define the following

radial interpolation:

R(t ) := (2−2|t |) · [|p|+εx]+ (2|t |−1) · [|p|−δ]

Then, for y with |y −q| ≥ 12εy , we define γ4 and γ5 as follows:

γ4(q + tc) := x : |x −G(y)| = R(t ),arg(p)−θx ≤ arg(x −G(y)) ≤ arg(p)

γ5(q + tc) :=

x : |x −G(y)| = R(t ),arg(p) ≤ arg(x −G(y)) ≤ arg(p)+θx

As R(t ) shrinks, γ2 and γ3 will shorten until they no longer are part of the contour. Whenthis happens, γ1 will partially intersect γ4 and γ5 until it moves completely out of the con-tour as well. We will show that the integrand is small along all parts of this contour, so thedetails of these intersections are not important.

This completes the description of a possible quasi-local contour near (p, q), but we willmorph it slightly so that it is more convenient. Consider applying the change of variablesgiven in Section 4.1. Since v = y , the v portion of the contour is identical to the y portionof the contour. Then, since u = x +χ1(v −q)+χ2(v −q)2, each contour γi (y) is translatedby χ1(v−q)+χ2(v−q)2, so that it retains its overall shape but is centered at a new location.


Using the chain rule, we compute the following:

∂H

∂u

∣∣∣∣(u,v)=(p,q)

= ∂H

∂x· ∂x

∂u+ ∂H

∂y· ∂y

∂u

∣∣∣∣(x,y)=(p,q)

= Hx(p, q)(3)

Since Hx(p, q) 6= 0 by assumption, and since H is analytic, the implicit function theoremguarantees that there exists a smooth parameterization κ(v) of the zero set of H , so thatH(p +κ(v), v) = 0 for v sufficiently close to q . Investigating κ(v) a little further, we use thepower series expansion of H about (p, q) to obtain the following:

0 = H(p +κ(v), v) = d10κ(v)+O(κ(v))2 +O(v −q)3

Thus, κ(v) = O(v −q)3. Additionally, the definitions of G(y), κ(v), and the change of vari-ables give us the following relation:

0 = H(p +G(y), y) = H(p +G(v)+χ1(v −q)+χ2(v −2)2, v)

However, we know already that κ(v) is the parameterization of the zero set of H . Thisforces the following relation:

G(y) = κ(y)−χ1(y −q)−χ2(y −q)2

Hence, G(y) = −χ1(y − q)−χ2(y − q)2 +O(v − q)3. This provides more insight into thischange of variables: in addition to allowing us to write H as a nice power series with somevanishing coefficients, the change of variables also describes V near (p, q). By convertingthe contour into (u, v)-coordinates, we are able to stabilize the u contours, slowing downthe movement of the zero set of H when it is parameterized by v . We take advantage ofthis slow-down by morphing our contour slightly, as described in the following paragraph.

In order to break the 2-dimensional Cauchy integral into two one-dimensional inte-grals, we need the quasi-local contour to be a product contour near the critical point,

(p, q). To achieve this goal, we will need to break into two cases: when |v − q| ≤ r− 25 and

when |v−q | > r− 25 . In the first case, |(v−q)3| ≤ r− 6

5 < 1r for r > 1. Hence, when |v−q | ≤ r− 2

5 ,

|κ(v)| = O(r− 65 ). Therefore, for r sufficiently large, the point p +κ(v) is always within the

circle of radius 1r about the point p, and we can morph our u-contour so that it is centered

exactly around the point p instead of the point p +κ(v). Thus, we will drop κ(v) from the

definitions of all the γi when |v −q | ≤ r− 25 , which means that the u contour no longer de-

pends on v when |v −q | ≤ r− 25 . (Note that this corresponds to a similar shift in the original

(x, y)-coordinates, which can be computed explicitly to justify that the original torus T can

be morphed locally to this new contour.) The portion of the contour where |v − q| ≤ r− 25

will yield the dominating contribution to the integral asymptotically.

In the other regime, when |v − q | ≥ r− 25 , we cannot simply eliminate κ(v). Instead, let

κ(v) be 0 when |v−q | ≤ r− 25 , let it be κ(v) when |v−q | ≥ r− 7

20 , and let it linearly interpolate

between 0 and κ(v) when r− 25 ≤ |v −q | ≤ r− 7

20 . We replace κ(v) with κ(v) in the definitionof the quasi-local cycle. Note that κ(v) =O(v3) as v tends to 0. This condition will be usedmuch later in the proof.

In summary, the final quasi-local cycle C (p, q) (in (u, v)-coordinates) near the criticalpoint (p, q) has three regimes. The contour is a line segment in v , and wraps around the

zero set of H in u. When |v − q| ≤ r− 25 , the u-contour wraps exactly around the point,

10 TORIN GREENWOOD

|q|

q

q - cε

q + cεy

y

Re y

Im y

Re x

Im x

p + G(y)

P(y)

FIGURE 3. The global y contour on the left, and the global x contour ξ(y)on the right, when x is away from p.

p, and this portion of the contour is a product contour. When r− 25 ≤ |v − q | ≤ εy

2 , the

contour instead wraps around the point, p+κ(v). Finally, if |v−q| ≥ εy

2 , then the u-contourgradually shrinks as |v −q| increases, until it no longer intersects the zero set of H at all.

4.3. Away from the Quasi-Local Cycle. Let us justify that away from this quasi-local cy-cle, the integrand decays exponentially. Consider the case where there is only one criticalpoint, (p, q). Beginning with the original torus T , if we shrink the x circle, then we canexpand the y circle (as discussed above) so that near q , it is a straight line segment with aspecified direction. From here, connect the two endpoints of the segment with a circulararc that has a radius that interpolates between the radii of the two endpoints of the linesegments. This y contour is pictured on the left in Figure 3.

For each y in this contour, there is a choice of |x| so that∣∣∣x y

1λ

∣∣∣= ∣∣∣pq1λ

∣∣∣. Call this choice

P (y). For each y in the contour above, consider the following arc:

ξ(y) = x : |x| = P (y), |arg(x −G(y))−arg(p)| ≥ θx

ξ(y) is pictured on the right in Figure 3.

Then, consider the following set of values:

R := (x, y)|y ∈ ζ, x ∈ ξ(y)

R is closed, and since (p, q) is assumed to be the only critical point, R also avoids the vari-ety V = (x, y)|H = 0 completely. Because the variety is also closed, R and V are separatedby open sets. Because R is compact, there is some number δ∗ > 0 small enough that boththe x and y portions of the contour above may be expanded outward by δ∗ and still not hitV . Thus, this section of the torus will have an integrand which decays exponentially fasterthan the integrand near the critical point.

The same is true for the closed set where y is on the arc described above with the linesegment omitted, and x is on the same circle as in ξ(y) but with |arg(x)−p| ≤ θx . This setis pictured in Figure 4.


|q|

q

q - cε

q + cεy

y

Re y

Im y

Re x

Im x

p + G(y)

P(y)

FIGURE 4. The global y contour on the left, and the global x contour on theright, when y is away from q .

After expanding the torus in the region away from the quasi-local cycle, we can con-nect the quasi-local cycle to the remaining part of the contour by an interpolation which

shrinks whichever contour has the larger value of∣∣∣x y

1λ

∣∣∣. If there is more than one minimal

critical point, we will add in more line segments through the y-values of the critical points,and we will interpolate between these line segments in a similar manner.

Hence, the asymptotic contribution to the integral comes from integrating over thequasi-local cycle C (p, q) described above.

5. A PRODUCT INTEGRAL

After applying the change of variables to the Cauchy integral formula (2) restricted tothe quasi-local cycle near (p, q), we obtain the following integral:(

1

2πi

)2 ÏC (p,q)

H(u, v)−β(u −χ1(v −q)−χ2(v −q)2)−r−1

v−s−1 du dv

Here, the Jacobian of the transformation is just 1:∣∣∣∣∣ ∂x∂u

∂x∂v

∂y∂u

∂y∂v

∣∣∣∣∣=∣∣∣∣1 −χ1 −2χ2(v −q)0 1

∣∣∣∣= 1

As mentioned above, our goal is to show that this integral is essentially a product integral.The following lemma describes this precisely.

Lemma 1.(1

2πi

)2 ÏC (p,q)

H(u, v)−β(u −χ1(v −q)−χ2(v −q)2)−r−1v−s−1 du dv

∼(

1

2πi

)2 ÏC`(p,q)

[Hx(p, q) ·(u−p)]−βu−r−1v−s−1[

1− χ1(v −q)+χ2(v −q)2

p

]−r−1

du dv

12 TORIN GREENWOOD

The above estimate holds as r, s →∞ withλ= r+O(1)s . Here, C`(p, q) is the portion of C (p, q)

where |v −q | ≤ r− 25 . Hence, C`(p, q) is a product contour.

The proof of this lemma will involve two types of statements: near the critical point,where |u −p| and |v − q | are both sufficiently small, we will argue that the integrands areasymptotically the same. Away from the critical point, where at least one of |u − p| or|v −q | is sufficiently large, we will show that both integrands are small, and hence do notcontribute asymptotically to either integral. (In the second integral, we need only showthat the integrand is small when |u −p| is large, since |v −q | is always small in C`(p, q).)

5.1. (u −p) and (v −q) are Small. In order to match the two integrands when |v −q| and|u −p| are small, we rewrite the original integrand first:

H(u, v)−β(u −χ1(v −q)−χ2(v −q)2)−r−1

v−s−1

= [Hx(p, q) · (u −p)]−βu−r−1v−s−1[

1− χ1(v −q)+χ2(v −q)2

p

]−r−1

K (u, v)L(u, v)

Here, K and L are correction factors with the following definitions:

K (u, v) :=1− χ1(v−q)+χ2(v−q)2

u

1− χ1(v−q)+χ2(v−q)2

p

r−1

L(u, v) :=[

H(u, v)

Hx(p, q)(u −p)

]−βThus, showing that the integrands in Lemma 1 are asymptotically equivalent is the sameas showing that K and L are asymptotically equal to 1. We will show this first for u ∈ γ1,and then for the parts of γ2 and γ3 sufficiently close to the critical point.

Lemma 2. For v ∈ [q − cεy , q + cεy ] with |v −q | ≤ r− 25 and u ∈ γ1, the following holds uni-

formly as r, s →∞ with λ= r+O(1)s :

K (u, v) = 1+o(1)

Proof. We pull aside the numerator of K (u, v):

1− χ1(v −q)+χ2(v −q)2

u= 1− χ1(v −q)+χ2(v −q)2

p − (p −u)

= 1− χ1(v −q)+χ2(v −q)2

p· 1

1−(1− u

p

)For u ∈ γ1, |u − p| = 1

r . Thus, we have(1− u

p

)= O

(r−1

), and

∣∣∣1− up

∣∣∣ < 1 for r sufficiently

large. Hence, we can expand 1

1−(1− u

p

) as a uniformly convergent geometric series for all

u ∈ γ1. This yields the following:

(4) 1− χ1(v −q)+χ2(v −q)2

u= 1− χ1(v −q)+χ2(v −q)2

p

[1+

(1− u

p

)+

(1− u

p

)2

+·· ·]


Now, we can replace the numerator in the base of K by the expression in (4) to obtain thefollowing:

(5)1− χ1(v−q)+χ2(v−q)2

u

1− χ1(v−q)+χ2(v−q)2

p

= 1−χ1(v−q)+χ2(v−q)2

p

1− χ1(v−q)+χ2(v−q)2

p

[(1− u

p

)+O

(1− u

p

)2]

Equation (5) holds uniformly for u ∈ γ1 and |v | ≤ r− 25 as r →∞. Plugging

(1− u

p

)=O

(r−1

)and v =O

(r− 2

5

)into (5) yields the following:

1− χ1(v−q)+χ2(v−q)2

u

1− χ1(v−q)+χ2(v−q)2

p

= 1+O(r− 7

5

)We replace the base of K with this new expression:

K (u, v) =(1+O

(r− 7

5

))−r−1 = e(−r−1)ln

(1+O

(r− 7

5

))

The Taylor series for the natural logarithm gives us the following estimate:

ln(1+O

(r− 7

5

))=O

(r− 7

5

)Thus, we may complete the lemma:

K (u, v) = e(−r−1)·O

(r− 7

5

)= e

O

(r− 2

5

)= 1+o(1)

Next, we prove the corresponding statement for L(u, v) on γ1.

Lemma 3. For v ∈ [q − cεy , q + cεy ] with |v −q | ≤ r− 25 and u ∈ γ1, the following holds uni-

formly as r, s →∞ with λ= r+O(1)s :

L(u, v) = 1+o(1)

Proof. Recall that H(u, v) has a particularly nice power series:

H(u, v) = ∑m,n≥0

dmn(u −p)m(v −q)n

In this series, we have the restrictions, d00 = d01 = d02 = 0. Hence, we can express H in thefollowing manner:

(6) H = d10(u −p)+ f (u, v)+ g (u, v)+h(u, v)

Here, we define f , g , and h by f (u, v) =O(u−p)2, g (u, v) =O((u −p)(v −q)

), and h(u, v) =

O(v − q)3, each uniformly as (u, v) approaches (p, q). Also, we note from (3) above thatd10 = Hx(p, q).

We now plug (6) into the definition of L:

L(u, v) :=[

H(u, v)

Hx(p, q)(u −p)

]−β=

[1+ f + g +h

Hx(p, q)(u −p)

]−β(7)

14 TORIN GREENWOOD

Using the fact that |u−p| = 1r for u ∈ γ1 and that |v −q | ≤ r− 2

5 gives the following relations:

f

Hx(p, q)(u −p)= O(u −p) =O

(r−1)

g

Hx(p, q)(u −p)= O(v −q) =O

(r− 2

5

)h

Hx(p, q)(u −p)= O

((v −q)3

u −p

)=O

(r ·

(r− 2

5

)3)=O

(r− 1

5

)Each of these statements holds uniformly for u ∈ γ1, |v−q| ≤ r− 2

5 , as r →∞. Plugging theserelations into (7) finishes the proof:

L(u, v) =[

1+O(r− 1

5

)]−β = 1+o(1)

We now turn to proving similar statements on the portions of γ2 and γ3 near the critical

point. Note that since κ(v) = 0 when |v − q| ≤ r− 25 , both γ2 and γ3 can be parameterized

by u(t ) = p + ωtr , with ω= e iφ and φ= arg p or arg p +2π.

Lemma 4. For v ∈ [q − cεy , q + cεy ] with |v −q | ≤ r− 25 , and for u ∈ γ2 ∪γ3 with u = p + ωt

r ,

t ≤ r3

10 , the following holds uniformly as r, s →∞ with λ= r+O(1)s :

K (u, v) = 1+o(1)

Proof. In this case, |v−q| =O(r− 2

5

)and |u−p| =O

(r

710

). We plug these into the expression

(5) for the base of K (u, v) (from the proof of Lemma 2):

1− χ1(v−q)+χ2(v−q)2

u

1− χ1(v−q)+χ2(v−q)2

p

= 1−χ1(v−q)+χ2(v−q)2

p

1− χ1(v−q)+χ2(v−q)2

p

[(1− u

p

)+O

(1− u

p

)2]

= 1+O(r− 2

5 · r− 710

)Plugging this into the definition of K (u, v) and using the Taylor series for the logarithmyields the bound we desire:

K (u, v) =(1+O

(r− 11

10

))−r−1

= e(−r−1)ln

(1+O

(r− 11

10

))

= e(−r−1)·O

(r− 11

10

)= 1+o(1)

Lemma 5. For v ∈ [q − cεy , q + cεy ] with |v −q | ≤ r− 25 , and for u ∈ γ2 ∪γ3 with u = p + ωt

r ,

t ≤ r3

10 , the following holds uniformly as r, s →∞ with λ= r+O(1)s :

L(u, v) = 1+o(1)


Proof. We borrow the expression (7) for L from Lemma 3:

L(u, v) =[

1+ f + g +h

Hx(p, q)(u −p)

]−βRecall that f , g , and h are defined by f (u, v) = O(u −p)2, g (u, v) = O

((u −p)(v −q)

), and

h(u, v) = O(v −q)3 as (u, v) approaches (p, q). In the region described in this Lemma, we

have the restrictions, 1r ≤ |u −p| ≤ r− 7

10 , and |v − q | ≤ r− 25 . Thus, we obtain the following

expressions:

f

Hx(p, q)(u −p)= O(u −p) =O

(r− 7

10

)g

Hx(p, q)(u −p)= O(v −q) =O

(r− 2

5

)h

Hx(p, q)(u −p)= O

((v −q)3

u −p

)=O

(r ·

(r− 2

5

)3)=O

(r− 1

5

)Plugging these into (7) above yields the desired result:

L(u, v) =[

1+O(r− 1

5

)]−β = 1+o(1)

This completes the proof that our integrand is essentially a product integrand near thecritical point. It remains to show that the contributions away from the critical point arenegligible.

5.2. (v −q) or (u−p) is Big. Here, we justify that the away from the critical point, the con-tribution to the integral decays exponentially faster than the contribution near the criticalpoint.

Lemma 6. Let C (p, q) represent the portion of C (p, q) where at least one of the following

conditions holds: |v − q| > r− 25 or |u − q | ≥ r− 7

10 . Then, the following holds uniformly asr, s →∞ with λ= r+O(1)

s :

(1

2πi

)2 ÏC (p,q)

H(u, v)−β(u −χ1(v −q)−χ2(v −q)2)−r−1

v−s−1 du dv

=O

(p−r q−srβe− 1

4|c| r15

)Proof. We bound the terms of the integrand separately. First, recall the nice power se-ries, H(u, v) = ∑

m,n≥0 dmn(u − p)m(v − q)n , with the relations, d00 = d01 = d02 = 0 andH(p +κ(v), v) = 0. Define u by u = u −p − κ(v) and v by v = v − q . H(p +κ(v), v) can berepresented as follows:

0 = H(p +κ(v), v) = d10κ(v)+d11κ(v)v +d20κ(v)2 +d03v3 +·· ·With this in mind, we expand the power series of H(p+κ(v)+u, v) to extract H(p+κ(v), v):

16 TORIN GREENWOOD

H(p + κ(v)+ u, v) = d10(κ(v)+ [κ−κ](v)+ u

)+d20(κ(v)+ [κ−κ](v)+ u

)2

+d11(κ(v)+ [κ−κ](v)+ u

) · (v)+d03v3 +·· ·

= H(p +κ(v), v)+d10([κ−κ](v)+ u

)+O

([κ−κ](v)

)2 +O(u)2 +O([κ−κ](v)u

)+O

(κ(v)[κ−κ](v)

)+O(κ(v)u

)+O

([κ−κ](v)v

)+O(uv

)To see that these seven big-O terms cover every possible term after the d03 term in theexpansion of H , notice that every term past the d03 term must have a power of (κ(v)+ [κ−κ](v)+u) of at least two, or a power of v of at least one and a power of (κ(v)+[κ−κ](v)+u)of at least one. Listing all of the terms of (κ(v)+[κ−κ](v)+u)2 and (κ(v)+[κ−κ](v)+u) · vand omitting the overlapping terms from H(p+κ(v), v) gives the seven big-O terms above.

Recall that [κ−κ](v) = O(v3

)and κ(v) = O

(v3

). For |v | ≤ r− 7

20 , this means that [κ−κ](v) =O

(r− 21

20

). However, for |v | ≥ r− 7

20 , κ is exactly κ. Thus, κ−κ=O(r− 21

20

). Additionally,

|u| ≥ 1r on all parts of C . Therefore, for εx , εy , and θx sufficiently small, all terms in the

expansion of H are negligible except d10u. Since |u| ≥ 1r , we have the following bound

uniformly on C when β> 0:

(8) H−β =O(rβ

)When β< 0, notice that H is bounded on compact sets, so H−β is bounded by a constant.Thus, regardless of the sign of β, we have the following bound:

H−β =O(r |β|

)Now, we turn to the remaining part of the integrand. Using the relation, s = r

λ+O(1) as

r, s →∞, we break down v−s−1:

v−s−1 = v−1+O(1)v− rλ

Now, we take a factor of p−r out of(u −χq (v −q)−χ2(v −q)2

)−rand a factor of q− r

λ out of

v− rλ to obtain the following decomposition:

(9)(u −χ1(v −q)−χ2(v −q)2)−r−1

v−s−1

= p−r q− rλ(u −χ1(v −q)−χ2(v −q)2)−1

v−1+O(1)e−rϕ(u,v)

Here, ϕ is defined by ϕ(u, v) = ln(

1p

[u −χ1(v −q)−χ2(v −q)2

])+λ−1 ln[

vq

]. We can ex-

pand ϕ as a bivariate power series and obtain the following:

(10) ϕ(u, v) = 1

p(u −p)+ M

2(v −q)2 +O

((u −p)(v −q)

)+O(u −p)2 +O(v −q)3

This equation holds uniformly as (u, v) approaches (p, q). M has the following definition:

M := ∂2ϕ

∂v2

∣∣∣∣(u,v)=(p,q)

=−2χ2

p− χ2

1

p2− 1

λq2


Recall the definition of c: c = M− 12 , so that M(v − q)2 = ( v

c

)2is always a nonnegative real

number. Then, we have:

ϕ(p + κ(v)+ u, q + cv) = 1

p(κ(v)+ u)+ 1

2

(v

c

)2

+O((κ(v)+ u)(v)

)+O(κ(v)+ u)2 +O(v)3

= 1

pu + 1

2

(v

c

)2

+O(uv)+O(u)2 +O(v)3

The last line holds uniformly as u, v → 0. From here, our goal is to bound e−rϕ in magni-tude. To do so, we will investigate the real part of ϕ. We break into cases here.

Case 1: When u ∈ γ1 and |v −q | ≥ r− 25 , we have that |u| = 1

r , which is much smaller than v2.

Re(ϕ(p + κ(v)+ u, q + cv)

)= Re

(1

2

(v

c

)2

+o(v2)

)≥ r− 4

5

4|c|The above inequality holds for r sufficiently large and for εx and εy small enough.Thus, we obtain the following for r sufficiently large:

∣∣e−rϕ(u,v)∣∣≤ e− r

15

4|c|

Case 2: Consider the case where u ∈ γ2 or γ3 and |u| ≤ r− 710 , but |v −q | ≥ r− 2

5 . (This case onlyapplies when |v − q| is sufficiently small, since γ2 and γ3 are not part of the contourif |v − q| is too large.) Here, we have that 1

p u = 1p (u − p −κ(v)) is a strictly positive

real number, and it dominates all u terms in ϕ(u, v). Thus, u only contributes to thedecay, and our bound is even more favorable than in Case 1. The following holds forsufficiently large r and sufficiently small εx and εy :


)≥ Re

(1

2

(v

c

)2

+o(v2)

)≥ r− 4

5

4|c|So, once again, we obtain this bound:

∣∣e−rϕ(u,v)∣∣≤ e− r

15

4|c|

Case 3: Consider the case where u ∈ γ2 or γ3 and |u| ≥ r− 310 . (Once again, this case is only

relevant when |v −q | is small enough for γ2 and γ3 to be part of the contour.) For suf-ficiently small εx and εy , the O(uv) term is dominated by the u term. The remaining

v terms are dominated by( v

c

)2, so these v terms can only increase the real part of ϕ.

Thus, the real part of ϕ is at least half the 1p u term, and we have the following for r

sufficiently large:


)≥ 1

2|p|r− 3

10

Plugging this into the exponential yields the following:∣∣e−rϕ(u,v)∣∣≤ e− 1

2|p| r7

10

Case 4: Now, consider the case where u ∈ γ4 or γ5 and |v−q | ≤ 12εy . Then, |u−κ(v)| = |p|+εx .

Thus, we have the following information about the leading term of ϕ:

18 TORIN GREENWOOD

κ(v)p

κ(v) ε

ε

θx

p1+

1+|p|x

x

|p|

up

α~ ~

FIGURE 5. α must be small when θx is small.

∣∣∣∣ 1

pu

∣∣∣∣ =∣∣∣∣ 1

p

∣∣∣∣ ∣∣u −p − κ(v)∣∣

≥∣∣∣∣ 1

p

∣∣∣∣[|u − κ(v)|− |p|]= εx

|p|(11)

Also, for θx sufficiently small (depending on εx and |p|), the following holds:

(12)∣∣arg(u −p − κ(v))−arg(p)

∣∣≤ π

3This statement should be clear graphically: let α = arg(u − p − κ(v)), and considerFigure 5. Clearly, as θx tends to zero, α approaches zero as well. Combining (11) and(12), we have the following:

Re

[1

pu

]≥ ε

|p| cos(arg(u −p − κ(v))−arg(p)

)≥ 1

2

εx

|p|Just like in Case 3, in the expansion of ϕ, the O(uv) term is dominated by the u term,

and the remaining v terms are dominated by( v

c

)2, which only adds to the real part of

ϕ. Hence, for εx ,εy , and θx sufficiently small we have:

Reϕ≥ 1

4

εx

|p|This yields: ∣∣e−rϕ(u,v)

∣∣≤ e− 14εx|p| r

This decay is much greater than in the other cases so far: this is because here, (u, v)is bounded away from (p, q) by a constant amount.

Case 5: Finally, consider the case where u ∈ γ4 or γ5, but |v − q | ≥ εy

2 . Here, both u and vare away from the critical point, (p, q). So, we expect the integrand to have the mostdecay here. However, there is some pain involved: namely, we must make sure thatour definition of δ still ensures that the integrand indeed decays.

Once again, the dominant terms in the expansion of ϕ are up and

( vc

)2. We check the

real component of up :

Re

(u

c

)= Re

(u

p−1− κ(v)

p

)= Re

(u − κ(v)

p

)−1(13)


Let us examine the remaining real part above. First, by the definitions of γ4 and γ5,the following is true:

arg(p)−θx ≤ arg(u − κ(v)) ≤ arg(p)+θx

Equivalently, we have the following:

−θx ≤ arg(u − κ(v))−arg(p) = arg

(u − κ(v)

p

)≤ θx

Therefore, we can bound the remaining real part in (13) as follows:

Re

[u − κ(v)

p

]≥

∣∣∣∣u − κ(v)

p

∣∣∣∣cosθx

≥(1− δ

|p|)

cosθx

In the last line, we used the fact that onγ4 andγ5, |u−κ(v)| ≥ |p|−δ. For θx sufficientlysmall, dependent on εy ,δ, and |p|, we can force this condition (assuming δ< |p|):

cosθx ≥ 1− δ

|p|−δRearranging, this gives us the following:(

1− δ

|p|)

cosθx ≥ 1− 2δ

|p|Plugging this into (13) yields:

Re

(u

p

)≥−2

δ

|p|

The( v

c

)2 term in the expansion ofϕ is real, and has a magnitude of at leastε2

y

4|c|2 (since

|v −q | ≥ εy

2 ). We impose the following restriction on δ:

δ< 1

2· |p|

2·ε2

y

4|c|2This gives us the following:

Re

(u

p

)+

(v

c

)2

≥ε2

y

8|c|2 ,

Finally, knowing that these two terms are the dominant terms of ϕ, we have the fol-lowing:

Re ϕ≥ε2

y

16|c|2Thus, we obtain exponential decay:

∣∣e−rϕ(u,v)∣∣≤ e

− ε2y

16|c|2 r

20 TORIN GREENWOOD

In every case, we have the following bound for εx ,εy ,θx , and δ sufficiently small and rsufficiently large:

(14)∣∣e−rϕ(u,v)

∣∣≤ e− r15

4|c|

Finally, notice that for εx , εy , θx , and δ sufficiently small,

(15)∣∣∣(u −χ1(v −q)−χ2(v −q)2)−1

v−1+O(1)∣∣∣≤ 2

∣∣p−1q−1+O(1)∣∣

Plugging (14) and (15) back into (9) gives the following:

(16)∣∣(u −χ1(v −q)−χ2(v −q)2)−r−1v−s−1

∣∣≤ 2p−r−1q−s−1+O(1)e− 12 r

15

Recognizing that the entire domain of integration has size bounded by a constant, wecombine (8) and (16) to get the desired result:(

1

2πi

)2 ÏC (p,q)

H(u, v)−β(u−χ1(v −q)−χ2(v −q)2)−r−1v−s−1 du dv =O

(p−r q−sr |β|e− 1

2 r15

)

Now we prove the corresponding statement for the product integral.

Lemma 7. Let C ∗`

represent the portion of C` where |u − q| ≥ r− 710 . Then, the following

holds uniformly as r, s →∞ with limr,s→∞ rs =λ:(

1

2πi

)2 ÏC ∗`

(p,q)[Hx(p, q) · (u −p)]−βu−r−1v−s−1

[1− χ1(v −q)+χ2(v −q)2

p

]−r−1

du dv

=O

(p−r q−sr |β|e− 1

4|c| r15

)Proof. This proof is a simplified version of the proof for Lemma 6.

First, notice that |u −p| ≥ 1r on C ∗

`. Hence, we have the following bound:

(17) [Hx(p, q) · (u −p)]−β =O(r |β|

)We manipulate the rest of the integrand:

(18) u−r−1v−s−1[

1− χ1(v −q)+χ2(v −q)2

p

]−r−1

= p−r q− rλ u−1v−1+O(1)

[1− χ1(v −q)+χ2(v −q)2

p

]−1

e−rϕ∗(u,v)

Here, ϕ∗(u, v) is defined by ϕ∗(u, v) = ln(

up

)+λ−1 ln

(vq

)+ ln

[1− χ1(v−q)+χ2(v−q)2

p

]. Ex-

panding ϕ∗(u, v) yields a power series very similar to the series for ϕ(u, v):

ϕ∗(u, v) = 1

p(u −p)+ M

2(v −q)2 +O

((u −p)(v −q)

)+O(u −p)2 +O(v −q)3

This equation holds uniformly as (u, v) approaches (p, q), and M has the same definitionas in Lemma 6. Notice that the first few terms of the power series of ϕ∗ match the termsof the power series of ϕ, (10), from Lemma 17. κ(v) is not present in C ∗

`, but we can


substitute 0 for κ(v) in the computations leading up to (14) in Lemma 6 (excluding theirrelevant cases where v −q is large), which shows that (14) still holds here:

(19)∣∣∣e−rϕ∗(u,v)

∣∣∣≤ e− 14|c| r

15

Finally, for εx ,εy ,θx , and δ sufficiently small, we have the following bound:

(20)

∣∣∣∣u−1v−1+O(1)[

1− χ1(v −q)+χ2(v −q)2

p

]−1

∣∣∣∣≤ 2∣∣p−1q−1+O(1)

∣∣Since the domain of integration has size bounded by a constant, combining (17), (19), and(20) finishes the proof.

We have nearly completed the proof of Lemma 1. However, it is not yet clear that thebounds we have found away from the critical point are small compared to the value ofthe whole integral. It remains to evaluate the asymptotic contribution of the product inte-gral, which will simultaneously show that the contributions to the integral away from thecritical point are negligible.

6. PROOF OF THEOREM

Lemma 1 has reduced our work to computing the following:(1

2πi

)2 ÏC`(p,q)

[Hx(p, q) · (u −p)]−βu−r−1v−s−1[

1− χ1(v −q)+χ2(v −q)2

p

]−r−1

du dv

We break it up into a product integral:(21)(

1

2πi

)2 (ˆU

[Hx(p, q) · (u −p)]−βu−r−1 du

)(ˆV

v−s−1[

1− χ1(v −q)+χ2(v −q)2

p

]−r−1

dv

)Above, U is the u-projection of the contour, C`, which resembles the x contour in Figure 2,

but with G(y) = 0. V is likewise the v-projection, which is the set,

v : v = q + ct , |t | ≤ r− 25

.

We analyze each of these two integrals separately.

Lemma 8. The following holds uniformly as r, s →∞ with λ= r+O(1)s :

ˆU

[Hx(p, q) · (u −p)]−βu−r−1 du

= 2πi

Γ(β)rβ−1p−r

(−Hx(p, q)p)−β

P

e−β(2πiω) +o(rβ−1p−r

)Here, ω is defined to be the signed number of times the curve H(t p, t q) crosses the branchcut in the definition of the function

x−β

P , as described in the statement of the Theorem.

Proof. The contour U is comprised of the segments γi for 1 ≤ i ≤ 5 in the case where |v −q | ≤ r− 2

5 . The endpoints of the contour, at the beginning of γ4 and end of γ5, both havemagnitude |u| = |p| + εx . We can attach these endpoints to a portion of the circle u :|u| = |p| + εx to form a closed cycle Cu that wraps around the origin and contains no

22 TORIN GREENWOOD

singularities of [Hx(p, q) · (u −p)]−β. Because u−r−1 is exponentially smaller on the circleu : |u| = |p|+εx than it is near the critical point p, we have:ˆ

U[Hx(p, q) · (u −p)]−βu−r−1 du = (1+o(1))

ˆCu

[Hx(p, q) · (u −p)]−βu−r−1 du

Now, we can use the Cauchy integral formula to evaluate this integral. However, we finallymust worry about how the analytic continuation of H−β is defined. H(0,0) is nonzero byassumption, and the values of H−β are defined near the origin of C2 by the generatingfunction itself. Separately from the analytic continuation of H−β that we have used up tothis point, we choose a branch of the logarithm with the following properties: the branchmust agree with H−β on some small neighborhood of the origin, and its branch cut mustbe a line from the origin that is not the line `(t ) = −t Hx(p, q)p for t ≥ 0, for any of thecritical points (p, q). Define

x−β

P as the value of x−β obtained by using this branch ofthe logarithm.

Consider the curve H(t p, t q) inC, with t ∈ [0,1). This curve may wrap around the originseveral times, and in particular, may cross the branch cut described above. Recall thebivariate power series for H(x, y):

H(x, y) = ∑m,n≥0

hmn(x −p)m(y −q)n

Plugging in our parameterization yields:

H(t p, t q) = h10(t p −p)+h01(t q −q)+·· ·= −(1− t )ph10 − (1− t )qh01 +O(1− t )2

= (1− t )(−ph10 −qh01)+O(1− t )2

The above equations are true as t → 1. Recall the following conditions: Hx(p, q) 6= 0, andHy (p, q) = p

λq Hx(p, q). Plugging this into our computations above yields:

H(t p, t q) = (1− t )(−p(1+λ)Hx(p, q))+O(1− t )2

Thus, as t tends to 1, the curve H(t q, t q) is essentially linear, with quadratic error. As longas the branch cut chosen above is not the line `(t ) mentioned above, the curve will onlycross the branch cut finitely many times. Let ω be the signed number of times the curveH(t p, t q) crosses the branch cut in the counter-clockwise direction for t ∈ [0,1). That is,every time the curve crosses the branch cut in the counter-clockwise direction, add 1 toω, and every time it crosses in the clockwise direction, subtract 1 from ω. If the curve onlytouches the branch cut without crossing it, leave ω unchanged.

As t approaches 1, we have shown that H behaves essentially like Hx(p, q)(u−p), and wehave traced how the argument changes as we expand the two-dimensional torus towardsthe critical point. Now, in order to revert the integral over Cu back to the appropriatecoefficient of Hx(p, q)(u − p) by using the Cauchy integral formula, we must follow theimage of Hx(p, q)(u−p) from u = p back to the origin u = 0. As u follows the line from p to0, the Hx(p, q)(u−p) will follow the line in C from 0 to −pHx(p, q), the point whose powerwe are trying to determine. Because this straight line is `(t ), it will not cross the branchcut we chose above. Thus, ω already accounts for the total number of times the branchcut is crossed. Figure 6 shows an example of this setup. In this example, ω = 1, becauseH(t p, t q) crosses the branch cut once in the counter-clockwise direction.


Re H

Im H

branch cut

H(0, 0)H(tp, tq)

-H (p, q)px

-tpH (p, q)x

FIGURE 6. An example with ω= 1.

In conclusion, we have the following:ˆCu

[Hx(p, q) · (u −p)]−βu−r−1 du = (1+o(1))2πi[ur ] (Hx(p, q) · (u −p))−β

= (1+o(1))2πi (Hx(p, q))r

(−βr

)·

(−Hx(p, q)p)−β−r

Pe−β(2πiω)(22)

From Stirling’s approximation, we have the following:(−βr

)∼ rβ−1

Γ(β)(−1)r

Additionally, we can separate the integer portion of the power of −Hx(p, q)p:(−Hx(p, q)p)−β−r

P=

(−Hx(p, q)p)−β

P

(−Hx(p, q)p)−r

Plugging these two expressions into (22) and simplifying yields the result:ˆ

Cu

[Hx(p, q) · (u −p)]−βu−r−1 du

= 2πi

Γ(β)rβ−1p−r

(−Hx(p, q)p)−β

P

e−β(2πiω) +o(rβ−1p−r

)

We turn our attention to the other integral, and find its asymptotic contribution.

Lemma 9. The following holds uniformly as r, s →∞ with λ= r+O(1)s :

ˆV

v−s−1[

1− χ1(v −q)+χ2(v −q)2

p

]−r−1

d v = cq−s−1

√2π

r+o

(q−s−1r− 1

2

)

24 TORIN GREENWOOD

Proof. Let v = q + cv , and apply this change of variables to the integral to obtain the fol-lowing:

ˆ[q−cεv ,q+cεv ]

v−s−1[

1− χ1

p(v −q)− χ2

p(v −q)2

]−r−1

dv

= cq−s−1ˆ

|v |≤r− 25

(1+ c

qv

)−s−1 [1− χ1

pcv − χ2

pc2v2

]−r−1

dv

We pull aside part of the integrand:(1+ c

qv

)=

(1+ c

qv

)− rλ+O(1)−1

The O(1) term is negligible: ∣∣∣∣∣(1+ c

qv

)O(1)∣∣∣∣∣ =

∣∣∣∣e log(1+ c

q v)O(1)

∣∣∣∣≤ e

C∣∣∣log

(1+ c

q v)∣∣∣

= 1+o(1)(23)

In the above equations, C is some real constant large enough to bound the O(1) term forall r . Using (23) yields the following:

cq−s−1ˆ

|v |≤r− 25

(1+ c

qv

)−s−1 [1− χ1

pcv − χ2

pc2v2

]−r−1

dv

= (1+o(1))cq−s−1ˆ

|v |≤r− 25

(1+ c

qv

)− rλ−1 [

1− χ1

pcv − χ2

pc2v2

]−r−1

dv

Define the following functions:

A(v) =(1+ c

qv

)·[

1− χ1

pcv − χ2

pc2v2

]ψ(v) = ln

[1− χ1

pcv − χ2

pc2v2

]+λ−1 ln

[1+ c

qv

]With these definitions, we obtain:(

1+ c

qv

)− rλ−1 [

1− χ1

pcv − χ2

pc2v2

]−r−1

= A(v)e−rψ(v)

From here, we can compute the following:

ψ(0) = 0

ψ′(0) = 0

ψ′′(0) = c2M = 1

A(0) = 1


Using standard Fourier-Laplace estimates gives the following for a constant εv > 0 suffi-ciently small:

(24)

ˆ εv

−εv

A(v)e−rψ(v) dv = (1+o(1))A(0)

√2π

ψ′′(0)re−rψ(0) = (1+o(1))

√2π

r

However, the domain of our integral is shrinking as r increases. To adjust for this, considerthe expansion for the phase function as r →∞ and |v |→ 0:

ψ(v) = v2 +O(v3)

Thus, for εy sufficiently small, if |v | ≥ r− 25 , we can bound the real part of the phase from

below:

Re(ψ(v)) ≥ r− 45

2

So, we can bound the exponential term in the integrand:

∣∣e−rψ(v)∣∣≤ e− r

15

2

Additionally, if εy is sufficiently small, then we can bound the amplitude, |A(v)| ≤ 2. Hence,

when |v | ≥ r− 25 with εy sufficiently small, the following holds:

∣∣A(v)e−rψ(v)∣∣≤ 2e− r

15

2

This gives us the following:

(25)

ˆr− 2

5 ≤|v |≤εv

A(v)e−rψ(v)dv =O

(e− r

15

2

)

Combining (24) and (25) completes the proof:

(1+o(1))cq−s−1ˆ

|v |≤r− 25

(1+ c

qv

)− rλ−1 [

1− χ1

pcv − χ2

pc2v2

]−r−1

dv

= cq−s−1

√2π

r+o

(q−s−1r− 1

2

)

Plugging the results of Lemma 8 and Lemma 9 into (21) gives us the final result:

(1

2πi

)2ˆU

[Hx(p, q) · (u −p)]−βu−r−1 du

ˆV

v−s−1[

1− χ1

p(v −q)− χ2

p(v −q)2

]−r−1

dv

=− i crβ−32(−Hx(p, q)p)−β

P e−β(2πiω)p−r q−s−1

Γ(β)p

2π

26 TORIN GREENWOOD

REFERENCES

[1] Y. Baryshnikov and R. Pemantle. Asymptotics of multivariate sequences, part III: quadratic points. InAdv. Math., vol 228 (2011) pp. 3127-3206. Available online:http://www.math.upenn.edu/~pemantle/papers/cones.pdf

[2] E.A. Bender and L.B. Richmond. Central and local limit theorems applied to asymptotic enumeration.II. Multivariate generating functions. In J. Combin. Theory Ser. A, vol 34 (1983) pp. 255-265.

[3] P. Flajolet and A. M. Odlyzko. Singularity analysis of generating functions. In SIAM J. Discrete Math., vol3 (1990) pp. 216-240. Available online:http://algo.inria.fr/flajolet/Publications/FlOd90b.pdf

[4] Z. Gao and L.B. Richmond. Central and local limit theorems applied to asymptotic enumeration. IV.Multivariate generating functions. In J. Comput. Appl. Math., vol 41 (1992) pp. 177-186.

[5] M. Goresky and R. MacPherson. Stratified Morse Theory, Springer-Verlag, Berlin Heidelberg, 1988. Avail-able online: http://www.math.ias.edu/~goresky/MathPubl.html

[6] Hwang, H.-K. Large deviations for combinatorial distributions. I. Central limit theorems. In Ann. Appl.Probab., vol 6 (1996) pp. 297-319. Available online:http://projecteuclid.org/download/pdf_1/euclid.aoap/1034968075

[7] Hwang, H.-K. Large deviations for combinatorial distributions. II. Local limit theorems. In Ann. Appl.Probab., vol 8 (1998) pp. 163-181. Available online:http://projecteuclid.org/download/pdf_1/euclid.aoap/1027961038

[8] R. Pemantle and M. Wilson. Asymptotics of multivariate sequences, part I: smooth points of the singularvariety. In J. Comb. Theory, Series A„ vol 97 (2002) pp. 129-161. Available online:http://www.math.upenn.edu/~pemantle/papers/p1.pdf

[9] R. Pemantle and M. Wilson. Asymptotics of multivariate sequences, part II: multiple points of the sin-gular variety. In Combinatorics, Probability and Computing, vol 13, no. 4 (2004) pp. 735-761. Availableonline: http://www.math.upenn.edu/~pemantle/papers/multiplepoints.pdf

[10] R. Pemantle and M. Wilson. Analytic Combinatorics in Several Variables, Cambridge University Press,New York, 2013. Available online: http://www.math.upenn.edu/~pemantle/ACSV.pdf

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF PENNSYLVANIA, 209 SOUTH 33RD STREET, PHILADEL-PHIA, PA 19104, USA

E-mail address: [email protected]

http://www.math.upenn.edu/~pemantle/papers/cones.pdf

http://algo.inria.fr/flajolet/Publications/FlOd90b.pdf

http://www.math.ias.edu/~goresky/MathPubl.html

http://projecteuclid.org/download/pdf_1/euclid.aoap/1034968075

http://projecteuclid.org/download/pdf_1/euclid.aoap/1027961038

http://www.math.upenn.edu/~pemantle/papers/p1.pdf

http://www.math.upenn.edu/~pemantle/papers/multiplepoints.pdf

http://www.math.upenn.edu/~pemantle/ACSV.pdf

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