On the Irreducibility of the Krawtchouck Polynomials

TL;DR

This paper investigates the irreducibility and Galois properties of Krawtchouck polynomials, revealing new cases of irreducibility and highlighting their complex Galois groups compared to other classical orthogonal polynomials.

Contribution

It determines the Newton Polygons of Krawtchouck polynomials and uncovers new irreducibility cases, advancing understanding of their algebraic properties.

Findings

Newton Polygons similar to Legendre polynomials

New cases of irreducibility identified

Galois groups are more complex than other classical families

Abstract

The Krawtchouck polynomials arise naturally in both coding theory and probability theory and have been studied extensively from these points of view. However, very little is known about their irreducibility and Galois properties. Just like many classical families of orthogonal polynomials (e.g. the Legendre and Laguerre), the Krawtchouck polynomials can be viewed as special cases of Jacobi polynomials. In this paper we determine the Newton Polygons of certain Krawtchouck polynomials and show that they are very similar to those of the Legendre polynomials (and exhibit new cases of irreducibility). However, we also show that their Galois groups are significantly more complicated to study, due to the nature of their coefficients, versus those of other classical orthogonal families.