Some irreducibility results for truncated binomial expansions

TL;DR

This paper proves that certain truncated binomial expansion polynomials are irreducible over the rationals within a specific range of parameters, contributing to algebraic understanding related to Schubert calculus.

Contribution

The paper establishes new irreducibility results for truncated binomial polynomials for a range of parameters, linking algebraic properties to geometric applications.

Findings

Polynomials $P_{n,k}(x)$ are irreducible over $\,\mathbb{Q}$ when $2\leq 2k \leq n < (k+1)^3$.

The irreducibility holds under the specified bounds, expanding the class of known irreducible polynomials.

Results have implications for algebraic geometry, particularly Schubert calculus in Grassmannians.

Abstract

For positive integers $n>k$, let $P_{n,k}(x)=\displaystyle\sum_{j=0}^k \binom{n}{j}x^j $ be the polynomial obtained by truncating the binomial expansion of $(1+x)^n$ at the $k^{th}$ stage. These polynomials arose in the investigation of Schubert calculus in Grassmannians. In this paper, the authors prove the irreducibility of $P_{n,k}(x)$ over the field of rational numbers when $2\leqslant 2k \leqslant n<(k+1)^3$.