| Résume | For 1 ≤ i ≤ n denote by $p_i=\sum_{j=1}^k the power-sums polynomials. For d ∈ N and y ∈ Rd the algebraic set Vd(y) = {x ∈ Rn : p1(x) = y1, . . . , pd(x) = yd} is called a Vandermonde-variety.
These algebraic sets have been studied by Arnold and Giventhal in connection to hyperbolic polynomials. In this talk we will generalise some results of Arnold and Giventhal on the Homology of Vandermonde varieties, more concretely: let H∗(Vd, Q) denote the cohomology group of a Vandermonde variety. Since Vd is invariant by the natural action of the symmetric group Sn the cohomology group H∗(Vd, Q) has the structure of an Sn module. We prove that for all λ ⊢ k, the multiplicity of the Specht-module Sλ in Hi(Vd(y), Q), is zero if length(λ) ≥ i+2d−3.
This vanishing result allows us to prove similar vanishing result for arbitrary symmetric semi- algebraic sets defined by symmetric polynomials of degrees bounded by d. As a result, we obtain for each fixed l ≥ 0, an algorithm for computing the first l + 1 Betti numbers of such sets, whose complexity is polynomially bounded (for fixed d and l). |
