A Unified Approach to Unimodality of Gaussian Polynomials

TL;DR

This paper introduces a computer algebra-based method to analyze the unimodality of Gaussian polynomials, providing a unified approach that extends previous results and proves new cases of related conjectures.

Contribution

It offers a novel algebraic approach using closed-form coefficients and cylindrical algebraic decomposition to study unimodality of Gaussian polynomials and related sequences.

Findings

Established a closed form for polynomial coefficients

Identified the exact range of strict unimodality

Proved two new cases of Stanley and Zanello's conjecture

Abstract

In 2013, Pak and Panova proved the strict unimodality property of $q$-binomial coefficients $\binom{\ell+m}{m}_q$ (as polynomials in $q$) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients. They showed it to be true for all $\ell,m\geq 8$ and a few other cases. We propose a different approach to this problem based on computer algebra, where we establish a closed form for the coefficients of these polynomials and then use cylindrical algebraic decomposition to identify exactly the range of coefficients where strict unimodality holds. This strategy allows us to tackle generalizations of the problem, e.g., to show unimodality with larger gaps or unimodality of related sequences. In particular, we present proofs of two additional cases of a conjecture by Stanley and Zanello.