Local $h^*$-Polynomials of Some Weighted Projective Spaces

TL;DR

This paper derives a general formula for local $h^*$-polynomials of certain weighted projective space simplices, proving their unimodality and revealing new real-rooted, symmetric Eulerian polynomials with interesting number-theoretic features.

Contribution

It provides a new explicit formula for local $h^*$-polynomials of specific lattice simplices and establishes their unimodality through real-rootedness, also discovering novel Eulerian polynomials.

Findings

Proved local $h^*$-polynomials are real-rooted and unimodal for certain simplices.

Derived a general formula for local $h^*$-polynomials of weighted projective space simplices.

Discovered a new class of real-rooted, symmetric Eulerian polynomials with number-theoretic properties.

Abstract

There is currently a growing interest in understanding which lattice simplices have unimodal local $h^\ast$-polynomials (sometimes called box polynomials); specifically in light of their potential applications to unimodality questions for Ehrhart $h^\ast$-polynomials. In this note, we compute a general form for the local $h^\ast$-polynomial of a well-studied family of lattice simplices whose associated toric varieties are weighted projective spaces. We then apply this formula to prove that certain such lattice simplices, whose combinatorics are naturally encoded using common systems of numeration, all have real-rooted, and thus unimodal, local $h^\ast$-polynomials. As a consequence, we discover a new restricted Eulerian polynomial that is real-rooted, symmetric, and admits intriguing number theoretic properties.