On the Irreducibility of a Truncated Binomial Expansion

Abstract

In the case that k = n− 1, the polynomial Pn,k(x) takes the form Pn,n−1(x) = (x+ 1) n − x. If n is not a prime, Pn,n−1(x) is reducible over Q. If n = p is prime, the polynomial Pn,n−1(x) = Pp,p−1(x) is irreducible as Eisenstein’s criterion applies to the reciprocal polynomial xPp,p−1(1/x). This note concerns the irreducibility of Pn,k(x) in the case where 1 ≤ k ≤ n− 2. Computations for n ≤ 100 suggest that in this case Pn,k(x) is always irreducible. We will not be able to establish this but instead give some results which give further evidence that these polynomials are irreducible. The problem arose during the 2004 MSRI program on “Topological aspects of real algebraic geometry” in the context of work of Inna Scherbak in her investigations of the Schubert calculus in Grassmannians. She had observed that the roots of any given Pn,k(x) are simple. This follows from the identity