Ultimate Positivity of Diagonals of Quasi-rational Functions

03/16/2017 ∙ by Hui Huang, et al. ∙ Johannes Kepler University Linz 0

The problem to decide whether a given multivariate (quasi-)rational function has only positive coefficients in its power series expansion has a long history. It dates back to Szego in 1933 who showed certain quasi-rational function to be positive, in the sense that all the series coefficients are positive, using an involved theory of special functions. In contrast to the simplicity of the statement, the method was surprisingly difficult. This dependency motivated further research for positivity of (quasi-)rational functions. More and more (quasi-)rational functions have been proven to be positive, and some of the proofs are even quite simple. However, there are also others whose positivity are still open conjectures. In this talk, we focus on a less difficult but also interesting question to decide whether the diagonal of a given quasi-rational function is ultimately positive, especially for the one conjectured to be positive by Kauers in 2007. To solve this question, it suffices to compute the asymptotics of the diagonal coefficients, which can be done by the multivariate singularity analysis developed by Baryshnikov, Pemantle and Wilson. Note that the ultimate positivity is a necessary condition for the positivity, and therefore can be used to either exclude the nonpositive cases or further support the conjectural positivity.

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1 Introduction

The problem to decide whether a given multivariate (quasi-)rational function has only positive coefficients in its power series expansion has a long history. It dates back to Szegö [10], who showed that for is positive, in the sense that all its series coefficients are positive, using an involved theory of special functions. In contrast to the simplicity of the statement, the method was surprisingly difficult. This dependency motivated further research for positivity of (quasi-)rational functions. More and more (quasi-)rational functions have been proven to be positive, and some of the proofs are even quite simple [4]. However, there are also others whose positivity are still open conjectures. For instance, the rational function with

is conjectured to be positive by Kauers [5], while no proof is available so far. This is equivalent to verify the positivity of the quasi-rational function for by [4, Proposition 1]. In this talk, we focus on a less difficult but also interesting question to decide whether the diagonal of is ultimately positive, inspired by [7, 9]. To solve this question, it suffices to compute the asymptotics of the diagonal coefficients, which can be done by the multivariate singularity analysis developed by Baryshnikov, Pemantle and Wilson [2, 8]. Note that the ultimate positivity is a necessary condition for the positivity, and therefore can be used to either exclude the nonpositive cases or further support the conjectural positivity.

2 Multivariate singularity analysis

In this section, we sketch and adapt the precise results for diagonals from [2, 8]. Reader interested in knowing more general cases may refer to the original text.

Let be a -variate complex generating function analytic at the origin, where , and . Then the diagonal of the rational function is defined to be . For simplicity, we assume to be a quasi-rational function of the form with a real number except for nonpositive integers and a polynomial. The zero set  of in is called the singular variety of . Note that since is analytic at the origin.

We aim to estimate the coefficient

asymptotically. Similar to the univariate case, we always start with the multivariate Cauchy integral formula

(1)

where is a sufficiently small torus around the origin and . The essential idea of the method in [2, 8] is to deform the contour without changing the integral (i.e. avoiding the points on ), such that the local behaviour of the integrand at so-called minimal critical points determines the asymptotics (under certain conditions). To describe minimal critical points, we need the definition of amoebas. Following [2], we let

Then the real set of for all is called the amoeba of the polynomial , denoted by . Note that amoebas can be computed effectively, see [11]. By [2, Proposition 2.2], there exists a component  of such that is convex and the set is precisely the open domain of convergence of the power series . Assume that there exists a unique point on the boundary minimizing the function with . We call the minimizing point for the diagonal. Let  denote the tangent cone to at , that is,

Let be the normal cone to

, namely the set of vectors

such that for all . Then [2, Definition 2.13] asserts that for each with there is a naturally defined cone (which is too lengthy to give here) that contains . We denote for the normal cone of and define the set of minimal critical points by

Note that . When is irreducible, for to be a smooth minimal critical point in the sense that the gradient of at is nonzero, we must have

(2)

We are mainly interested in the following quadratic case.

Theorem 1 ([2, Proposition 3.7]).

Let be a -variate quasi-rational function with and a polynomial. Let be a component of so that has a convergent power series expension in . Assume that there exists a minimizing point for the diagonal, and the set contains only one point . Further assume that the leading homogeneous part of at with is an irreducible quadratic with the matrix congruent to the diagonal matrix . Then, when the Gamma functions in the denominator are finite,

(3)

where is the dual quadratic form of with the matrix .

3 Asymptotics of diagonals

In this section, we apply the multivariate singularity analysis to two quasi-rational functions. The first example comes from a well-known rational function, which was shown to be positive for in [10, 1].

Example 2.

Consider the quasi-rational function

We are interested in the asymptotics for the diagonal coefficents of . For simplicity, we tranlate each coordinate to , and then apply the method to with

Identify minimal critical points. Let be zero set of . It is readily seen that is smooth except for the point . Let be the component of corresponding to the power series expension of  at the origin. Then contains the negative orthant by [11]. We claim that is on the boundary of . Indeed, it suffices to verify that is nonzero in the open unit polydisk . Following [2, Section 4.4], it is equivalent to show that is nonzero in the open disk by sending to . Then further setting to changes the problem to prove that

is nonzero in , which is trivial since . Since the diagonal direction , the point is the minimizing point for the diagonal by the definition of normal cones. At the point , the leading homogeneous term of composing with the exponential is

with the matrix congruent to the diagonal matrix . Then the dual quadratic form of is

By definition, the normal cone is the set containing the diagonal direction . Hence the point . The remaining points in could only be smooth critical points on . Solving (2) implies that has only one point, namely . To get the leading term of the asymptotics, it suffices to compute the contribution from .

Compute diagonal asymptotics. So far, we have shown that satisfies the hypotheses in Theorem 1. Since , by (3) the -th diagonal coefficient of is asymptotic to

(4)

which implies that the diagonal of is ultimately positive for .

Now let’s turn to the function that we mention in the introduction.

Example 3 ([5]).

Consider the quasi-rational function

with . We want to know the diagonal asymptotics of . Similar to the previous example, we first translate each coordinate to and work with where

Then the diagonal asymptotics of can be easily computed.

Identify minimal critical points. Let be the zero set of . Then the only non-smooth point on is the point . Again the component of corresponding to the power series expansion of  at the origin is the one contains the negative orthant. Again, we claim that is on the boundary of . Similarly, it suffices to verify that is nonzero in the open unit polydisk , which is then equivalent to show that the numerator of

is nonzero in . This can be done by cylindrical algebraic decomposition (CAD) [3, 6] as follows. Suppose that is a zero of the numerator. Then we can represent as

whose denominator cannot be zero since for . Applying CAD shows that the real part of , in terms of real and imaginary parts of , must be greater than , a contridiction. Since the diagonal direction belongs to , the point is the minimizing point for the diagonal by the definition of normal cones. At the point , the leading homogeneous term of is

with the matrix

congruent to the diagonal matrix . Then the dual quadratic form is

Then the normal cone By the same reason as the previous example, the set only contains the point , which determines the leading term of the asymptotics.

Computing diagonal asymptotics. We have seen that satisfies the hypotheses in Theorem 1. Hence by (3), the -th diagonal coefficient of is asymptotic to

which implies that the diagonal of is ultimately positive for .

Remark 4.

For the cases when the Gamma functions in the denominator of (3) is infinite, e.g., when for Example 2 and for Example 3, we need more techniques from [8] to compute the diagonal asymptotics, which will be addressed in future work.

4 Acknowledgement

I would like to thank my advisor Manuel Kauers for encouraging me to work with this topic and also providing valuable comments.

References

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