Laurent polynomials, Eulerian numbers, and Bernstein's theorem

TL;DR

This paper provides an alternative proof connecting Laurent polynomials, Eulerian numbers, and Bernstein's theorem, revealing new combinatorial interpretations and clarifying their mathematical relationships.

Contribution

It offers a new proof of known results linking Laurent polynomials and Eulerian numbers using Bernstein's theorem, and introduces a combinatorial interpretation of certain hypercube volumes.

Findings

Expected number of specific Laurent polynomials equals an Eulerian number

Refinement of Eulerian numbers relates to hypercube hyperplane sections

Bernstein's theorem clarifies connections between algebraic and combinatorial objects

Abstract

Erman, Smith, and V\'arilly-Alvarado showed that the expected number of doubly monic Laurent polynomials $f(z) = z^{-m} + a_{-m+1}z^{-m+1} + \cdots + a_{n-1}z^{n-1} + z^n$ whose first $m+n-1$ powers have vanishing constant term is the Eulerian number $\brac{m+n-1}{m-1}$, as well as a more refined result about sparse Laurent polynomials. We give an alternate proof of these results using Bernstein's theorem that clarifies the connection between these objects. In the process, we show that a refinement of Eulerian numbers gives a combinatorial interpretation for volumes of certain rational hyperplane sections of the hypercube.