The homotopy type of spaces of resultants of bounded multiplicity

2016/12/20 by Kozlowski, Andrzej, Yamaguchi, Kohhei
#55P10 #55P35 #55R80 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1612.06793

Abstract

For positive integers m,n, d≥ 1 with (m,n)\not= (1,1) and a field \Bbb F with its algebraic closure \Bbb F, let Poly^d,m_n(\Bbb F) denote the space of all m-tuples (f₁(z),⋯ ,f_m(z))∈ \Bbb F [z] of monic polynomials of the same degree d such that polynomials f₁(z),⋯ ,f_m(z) have no common root in \Bbb F of multiplicity ≥ n. These spaces were defined by Farb and Wolfson in \citeFW as generalizations of spaces first studied by Arnold, Vassiliev, Segal and others in different contexts. In \citeFW they obtained algebraic geometrical and arithmetic results about the topology of these spaces. In this paper we investigate the homotopy type of these spaces for the case \Bbb F =ℂ. Our results generalize those of \citeFW for \Bbb F =\Bbb C and also results of G. Segal \citeSe, V. Vassiliev \citeVa and F.Cohen-R.Cohen-B.Mann-R.Milgram \citeCCMM for m≥ 2 and n≥ 2.

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