Unimodality and certain bivariate formal Laurent series

TL;DR

This paper investigates the unimodality of specific bivariate Laurent series, demonstrating their algebraic structure and applying these results to prove unimodality in various combinatorial and geometric contexts.

Contribution

It establishes the semiring structure of certain Laurent series and uses this to prove new unimodality results in partition theory, Gauss polynomials, and algebraic geometry.

Findings

Proves the semiring structure of formal bivariate Laurent series with non-negative coefficients.

Solves an open problem by Andrews on the unimodality of generalized Gauss polynomials.

Establishes unimodality of Betti numbers and Gromov-Witten invariants in specific Hilbert schemes.

Abstract

In this paper, we examine the unimodality and strict unimodality of certain formal bivariate Laurent series with non-negative coefficients. We show that the sets of these formal bivariate Laurent series form commutative semirings under the operations of addition and multiplication of formal Laurent series. This result is used to establish the unimodality of sequences involving Gauss polynomials and certain refined color partitions. In particular, we solve an open problem posed by Andrews on the unimodality of generalized Gauss polynomials and establish an unimodal result for a statistic of plane partitions. We also establish many unimodal results for rank statistics in partition theory, including the rank statistics of concave and convex compositions studied by Andrews, as well as certain unimodal sequences studied by Kim-Lim-Lovejoy. Additionally, we establish the unimodality of the Betti numbers and Gromov-Witten invariants of certain Hilbert schemes of points.