Spaces of polynomials with constrained real divisors, II. (Co)homology & stabilization

TL;DR

This paper studies the topology of spaces of real polynomials with constrained root multiplicities, providing combinatorial methods for homology computation and stabilization results as degree increases.

Contribution

It introduces a combinatorial approach to compute homology of polynomial spaces with constrained divisors and establishes stabilization results for their cohomology as degree grows.

Findings

Homology of polynomial spaces reduced to combinatorial differential complexes

Stabilization of cohomology as polynomial degree tends to infinity

Explicit computation of homology for discriminants of binary forms

Abstract

In the late 80s, V.~Arnold and V.~Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree which have no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces P^{c\Theta}_d of real monic univariate polynomials of degree d whose real divisors avoid given sequences of root multiplicities. These forbidden sequences are taken from an arbitrary poset \Theta of compositions that are closed under certain natural combinatorial operations. We reduce the computation of the homology H_*(P^{c\Theta}_d) to the computation of the homology of a differential complex, defined purely combinatorially in terms of the given closed poset \Theta. We also obtain the stabilization results about H^\ast(P^{c \Theta}_d), as d goes to infinity. These results are deduced from our description of the homology of spaces B^{c \Theta}_d whose points are binary real homogeneous forms, considered up to projective equivalence, with similarly \Theta-constrained real divisors. In particular, we exhibit differential complexes that calculate the homology of these spaces and obtain some stabilization results for H^*(B^{c \Theta}_d), as d goes to infinity. In particular, we compute the homology of the discriminants of projectivized binary real forms for which there is at least one line on which the form vanishes with multiplicity >= 2 and of their complements in \cB_d \cong RP^d.