Real-Rootedness and Gamma-Positivity for a Variation of the Morris Constant Term
Abstract
Beck and Pixton expressed the Ehrhart polynomial of the Birkhoff polytope as a weighted sum of constant terms of several multivariate rational functions. Xin and Zhang studied a class of constant terms hn(t), which can be regarded as a variation of the Morris constant term. They proved that hn(t) is a polynomial of degree (n−1)2 and obtained many nice properties involving the Morris constant term identity. Let hn*(y)=(1−y)(n−1)2+1∑t≥0hn(t)yt. For fixed n≥ 3, we obtain the following results for hn*(y): (i): hn*(y) is a polynomial with positive integer coefficients. (ii): hn*(y) is real-rooted. In particular, all its roots are non-positive real numbers. (iii): hn*(y) is Gamma-positive. Furthermore, hn*(y) is palindromic, unimodal, and ultra log-concave. This confirms Xin and Zhang's conjecture regarding hn*(y). In order to resolve this conjecture, we also developed an operator on the space of symmetric polynomials that preserves real stability. As a byproduct, we prove that every root of a Gamma polynomial associated with hn*(y) is a negative real number.
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BibTeX
@misc{liu2026realrootedness,
title = {{Real-Rootedness and Gamma-Positivity for a Variation of the Morris Constant Term}},
author = {Feihu Liu and Zihao Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.15201},
eprint = {2609.15201},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15201},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Real-Rootedness and Gamma-Positivity for a Variation of the Morris Constant Term
AU - Feihu Liu
AU - Zihao Zhang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2609.15201
UR - https://arxiv.org/abs/2609.15201
AB - Beck and Pixton expressed the Ehrhart polynomial of the Birkhoff polytope as a weighted sum of constant terms of several multivariate rational functions. Xin and Zhang studied a class of constant terms h_n(t), which can be regarded as a variation of the Morris constant term. They proved that h_n(t) is a polynomial of degree (n−1)^2 and obtained many nice properties involving the Morris constant term identity. Let h_n^*(y)=(1−y)^(n−1)^2+1∑_t≥0h_n(t)y^t. For fixed n≥ 3, we obtain the following results for h_n^*(y): (i): h_n^*(y) is a polynomial with positive integer coefficients. (ii): h_n^*(y) is real-rooted. In particular, all its roots are non-positive real numbers. (iii): h_n^*(y) is Gamma-positive. Furthermore, h_n^*(y) is palindromic, unimodal, and ultra log-concave. This confirms Xin and Zhang's conjecture regarding h_n^*(y). In order to resolve this conjecture, we also developed an operator on the space of symmetric polynomials that preserves real stability. As a byproduct, we prove that every root of a Gamma polynomial associated with h_n^*(y) is a negative real number.
ER -