Real polynomials with constrained real divisors. I. Fundamental groups

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 [Formula: see text] and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies, we consider the spaces [Formula: see text] of real monic univariate polynomials of degree [Formula: see text] whose real divisors avoid sequences of root multiplicities, taken from a given poset [Formula: see text] of compositions which is closed under certain natural combinatorial operations. In this paper, we concentrate on the fundamental group of [Formula: see text] and of some related topological spaces. We find explicit presentations for the groups [Formula: see text] 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 [Formula: see text]. We also show that [Formula: see text] stabilizes for [Formula: see text] large. The mechanism that generates [Formula: see text] has similarities with the presentation of the braid group as the fundamental group of the space of complex monic degree [Formula: see text] polynomials with no multiple roots and with the presentation of the fundamental group of certain ordered configuration spaces over the reals which appear in the work of Khovanov. We further show that the groups [Formula: see text] admit an interpretation as special bordisms of immersions of one-manifolds into the cylinder [Formula: see text], whose images avoid the tangency patterns from [Formula: see text] with respect to the generators of the cylinder.