Year: 2013
Type: preprint
Abstract: Erman, Smith, and Várilly-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.
Source: arXiv (Cornell University)
Author: Liu, Ricky Ini
Institution: University of Michigan
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W2951894079