Coincident root loci and Jack and Macdonald polynomials for special values of the parameters

TL;DR

This paper studies the algebraic structure of polynomials with multiple roots using Jack and Macdonald polynomials at special parameters, providing explicit bases, formulas, and generalizations.

Contribution

It introduces a basis for the ideal of polynomials with double roots using Jack polynomials at a specific parameter value, and extends results to Macdonald and interpolation Jack polynomials.

Findings

Explicit basis for the ideal using Jack polynomials at alpha = -2

Formula for the Hilbert-Poincaré series of the ideal

Generalization to Macdonald and interpolation Jack polynomials

Abstract

We consider the coincident root loci consisting of the polynomials with at least two double roots andpresent a linear basis of the corresponding ideal in the algebra of symmetric polynomials in terms of the Jack polynomials with special value of parameter $\alpha = -2.$ As a corollary we present an explicit formula for the Hilbert-Poincar\`e series of this ideal and the generator of the minimal degree as a special Jack polynomial. A generalization to the case of the symmetric polynomials vanishing on the double shifted diagonals and the Macdonald polynomials specialized at $t^2 q = 1$ is also presented. We also give similar results for the interpolation Jack polynomials.