Computing critical points for algebraic systems defined by hyperoctahedral invariant polynomials
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Let π be a field of characteristic zero and π[x_1, β¦, x_n] the corresponding multivariate polynomial ring. Given a sequence of s polynomials π = (f_1, β¦, f_s) and a polynomial Ο, all in π[x_1, β¦, x_n] with s<n, we consider the problem of computing the set W(Ο, π) of points at which π vanishes and the Jacobian matrix of π, Ο with respect to x_1, β¦, 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(Ο, π) according to the orbits of B_n and then design an algorithm whose output is a hyperoctahedral representation of W(Ο, π). The runtime of our algorithm is polynomial in the total number of points described by the output.
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