Homotopy stability for spaces of triples of real polynomials without common roots

Abstract

We continue our study of the topology of the spaces of $m$ tuples of real polynomials with common degree $d$ and without common roots of multiplicity $n$, and in particular their stability properties with respect to $d$. In an earlier paper we have proved a homotopy stability result and determined the stable homotopy types of such spaces in the case $m n >=4$. In the case $m n= 3$ we could only prove stability in homology. In this paper we settle the case (m,n) = (3,1): the space of triples of monic real polynomials of the same degree having no common root. We prove homotopy stability and determine the precise stability range supplied by our method. We also show that in degree 2, the distinction between homotopy equivalence up to and through the stability dimension is essential. The other borderline case (m,n) = (1,3) is not treated here. We expect that homotopy stability will hold also in that case and hope to return to it in future work.