Spaces of nonresultant systems of bounded multiplicity with real coefficients

For each pair .m;n/ of positive integers with .m;n/ ¤ .1;1/ and an arbitrary field F with algebraic closure F , let Poly d;m n .F / denote the space of m-tuples .f 1 .z/;: : : ; f m .z// 2 F Œz m of monic polynomials with coefficients in F of the same degree d such that the polynomials ff k .z/gm kD1 have no common root in F of multiplicity n.These spaces Poly d;m n .F / were first defined and studied by B. Farb and J. Wolfson as generalizations of spaces first studied by Arnold, Vassiliev and Segal and others in several different contexts.In a previous paper we determined explicitly the homotopy type of this space when F D C.Here we investigate this space when F D R. 1 Introduction Motivation We build on the works of V. I. Arnold [1], V. Vassiliev [22], G. Segal [17], and B. Farb and J. Wolfson [4].Arnold considered the space SP d n .C/ of complex-coefficient monic polynomials of degree d without roots of multiplicity n, which plays a role in the theory of singularities.For example, if n D 2, this space is the same as the space of complex monic polynomials of degree d without repeated roots, and this is homotopy equivalent to the Eilenberg-Mac Lane space K.Br.d/; 1/, where Br.d/ denotes the Artin braid group of d strings.Arnold [1] computed the homology of these spaces and established their homology stability.His results were much extended and generalized by Vassiliev [22].Analogous results were discovered by Segal [17] in a different context inspired by control theory.Segal considered the space Hol d .S 2 ; CP m 1 / of basepoint-preserving holomorphic maps of degree d from the Riemann sphere S 2 to the .m1/-dimensional complex projective space CP m 1 , and its inclusion into the space Map d .S 2 ; CP m 1 / D 2 d CP m 1 of corresponding basepoint-preserving continuous maps.Intuitive considerations based on Morse theory suggest that the homotopy type of the first space should approximate that of the second space more and more closely as the degree d increases.Segal proved that this is true by observing that the space Hol d .S 2 ; CP m 1 / can be identified with the space of m-tuples .f 1 .z/;: : : ; f m .z// 2 CŒz m of monic polynomials of the same degree d without common roots.He defined a stabilization map Hol d .S 2 ; CP m 1 / !Hol d C1 .S 2 ; CP m 1 / and proved that the induced maps on homotopy groups are isomorphisms up to some dimension increasing with d.Using a different technique he also proved that there is a homotopy equivalence S W lim !Hol d .S 2 ; CP m 1 / ' !