On some topological invariants of algebraic functions associated to the Young stratification of polynomials. (Q1412700)

Let \(\text{ P}^n\) denote the space consisting of all monic complex polynomials of degree \(n\) and \(\Sigma=\Sigma^n(m_1,\cdots ,m_k)\subset \text{ P}^n\) the subspace of monic polynomials having roots of multiplicities at least \(m_1,m_2,\cdots ,m_k\), where \(m_1\geq m_2\geq \cdots \geq m_k\geq 2\) are fixed integers. In this paper the author studies the homology \(H_*(\text{ P}^n\Sigma)\) of the complement \(\text{ P}^n\Sigma=\text{ P}^n-\Sigma\). In particular, he proves that they satisfy three theorems: the stabilization theorem (i.e. they stabilize if the degree of the polynomials tends to infinity), the repetition theorem (i.e. homology groups associated to spaces of polynomials of successive even and odd degrees are isomorphic) and the finiteness theorem (all cohomology groups are finite except for two degrees). These three theorems generalize the classical basic theorems obtained by \textit{V. I. Arnol'd} in his famous paper on topological invariants of algebraic functions [Trans. Mosc. Math. Soc. 21 (1970), 30-52 (1971; Zbl 0225.14005)].