Real polynomials with constrained real divisors. I. Fundamental groups

TL;DR

This paper studies the fundamental groups of spaces of real polynomials with constrained root multiplicities, providing explicit presentations and showing stabilization and freeness properties in certain cases.

Contribution

It introduces explicit presentations for the fundamental groups of these polynomial spaces and demonstrates their stabilization and freeness under specific conditions.

Findings

Fundamental groups are explicitly presented with generators and relations.

In many cases, these groups are free with rank bounded quadratically in degree d.

The fundamental groups stabilize for large degree d.

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 d and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces of real monic univariate polynomials of degree d whose real divisors avoid sequences of root multiplicities taken from a given poset of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of such spaces. We find explicit presentations for the fundamental groups in terms of generators and relations and show that in a number of cases they are free with rank bounded from above by a quadratic function in d. We also show that the fundamental group stabilizes for d large. We further show that the fundamental groups admit an interpretation as special bordisms of immersions of 1-manifolds into the cylinder S^1 \times R, whose images avoid the tangency patterns from the poset with respect to the generators of the cylinder.