# Ultimate Positivity of Diagonals of Quasi-rational Functions

## TL;DR

This paper investigates the ultimate positivity of diagonals of multivariate quasi-rational functions, using asymptotic analysis to support or refute conjectures about their positivity, especially focusing on a conjecture by Kauers.

## Contribution

It applies multivariate singularity analysis to determine the asymptotic behavior of diagonals, providing a method to assess positivity conjectures for quasi-rational functions.

## Key findings

- Identifies cases where diagonals are ultimately positive or not.
- Supports Kauers' conjecture through asymptotic analysis.
- Provides a computational approach for positivity testing.

## Abstract

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.