Computing critical points for algebraic systems defined by hyperoctahedral invariant polynomials
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Let 𝕂 be a field of characteristic zero and 𝕂[x_1, …, x_n] the corresponding multivariate polynomial ring. Given a sequence of s polynomials 𝐟 = (f_1, …, f_s) and a polynomial ϕ, all in 𝕂[x_1, …, x_n] with s<n, we consider the problem of computing the set W(ϕ, 𝐟) of points at which 𝐟 vanishes and the Jacobian matrix of 𝐟, ϕ with respect to x_1, …, x_n does not have full rank. This problem plays an essential role in many application areas. In this paper we focus on a case where the polynomials are all invariant under the action of the signed symmetric group B_n. We introduce a notion called hyperoctahedral representation to describe B_n-invariant sets. We study the invariance properties of the input polynomials to split W(ϕ, 𝐟) according to the orbits of B_n and then design an algorithm whose output is a hyperoctahedral representation of W(ϕ, 𝐟). The runtime of our algorithm is polynomial in the total number of points described by the output.
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