Combinatorics and Topology of Spaces of Real Polynomials

A seminar to be conducted by Dr. Volkmar Welker, Professor at Philips University of Marburg, Germany.

Abstract

In this talk, I survey recent results (from joint research with Gabriel Katz and Boris Shapiro) about spaces of real polynomials of fixed degree n and restrictions on root multiplicities. Our restrictions on root multiplicities can be expressed through integer partitions, e.g., the partition (2; 2; 1) of 5 would say that we only consider real polynomial of degree n, which has no 5 roots that are all real, and for which two pairs of them coincide. We study a stratification of these spaces by cells indexed by compositions of integers (i.e., ordered number partitions) and show that the cellular differential of the corresponding CW-complex has a simple combinatorial description. From these facts, we deduce results on the fundamental group and the homology groups of our spaces. In particular, we consider the limiting behavior as the degree n goes to infinity.

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