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

M. Kasatani·T. Miwa·A. N. Sergeev·A. P. Veselov
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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 $α= -2.$ As a corollary we present an explicit formula for the Hilbert-Poincarè 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.

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M. Kasatani, T. Miwa, A. N. Sergeev, A. P. Veselov. 2004-04-05. Coincident root loci and Jack and Macdonald polynomials for special values of the parameters. https://arxiv.org/abs/math/0404079

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