Weighted Ehrhart Theory: Extending Stanley's nonnegativity theorem

Authors

Esme Bajo
Robert Davis
Jesús A. De Loera
Alexey Garber, The University of Texas Rio Grande Valley
Sofía Garzón Mora
Katharina Jochemko
Josephine Yu

Document Type

Article

Publication Date

5-2024

Abstract

We generalize R. P. Stanley's celebrated theorem that the h∗-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h∗-polynomial as a real-valued function for a larger family of weights.

We generalize R. P. Stanley's celebrated theorem that the h ⁎ -polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h ⁎ -polynomial as a real-valued function for a larger family of weights.

We explore the case when the weight function is the square of a single (arbitrary) linear form. We show stronger results for two-dimensional convex lattice polygons and give concrete examples showing tightness of the hypotheses. As an application, we construct a counterexample to a conjecture by Berg, Jochemko, and Silverstein on Ehrhart tensor polynomials.

Comments

Original published version available at https://doi.org/10.1016/j.aim.2024.109627

Recommended Citation

Bajo, Esme, Robert Davis, Jesús A. De Loera, Alexey Garber, Sofía Garzón Mora, Katharina Jochemko, and Josephine Yu. "Weighted Ehrhart theory: Extending Stanley's nonnegativity theorem." Advances in Mathematics 444 (2024): 109627. https://doi.org/10.1016/j.aim.2024.109627

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Publication Title

Advances in Mathematics

DOI

10.1016/j.aim.2024.109627