Laurent polynomials, Eulerian numbers, and Bernsteinʹs theorem

Erman, Smith, and Várilly-Alvarado (2011) showed that the expected number of doubly monic Laurent polynomials f ( z ) = z − m + a − m + 1 z − m + 1 + ⋯ + a n − 1 z n − 1 + z n whose first m + n − 1 powers have vanishing constant term is the Eulerian number 〈 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 Bernsteinʹs 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.