The homology of spaces of polynomials with roots of bounded multiplicity

TL;DR

This paper computes the integral homology groups of spaces of monic complex polynomials with degree k and at most l roots of multiplicity n, revealing their topological structure.

Contribution

It provides the first explicit calculation of the integral homology groups for these polynomial spaces with bounded root multiplicity.

Findings

Homology groups of P_{k, n}^l are explicitly determined.

Results extend understanding of polynomial root space topology.

New formulas for homology groups depending on k, n, l.

Abstract

Let P_{k, n}^l be the space consisting of monic complex polynomials f(z) of degree k and such that the number of n-fold roots of f(z) is at most l. In this paper, we determine the integral homology groups of P_{k, n}^l.