Computing critical points for invariant algebraic systems
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Let 𝐊 be a field and ϕ, 𝐟 = (f_1, …, f_s) in 𝐊[x_1, …, x_n] be multivariate polynomials (with s < n) invariant under the action of 𝒮_n, the group of permutations of {1, …, n}. We consider the problem of computing the points at which 𝐟 vanish and the Jacobian matrix associated to 𝐟, ϕ is rank deficient provided that this set is finite. We exploit the invariance properties of the input to split the solution space according to the orbits of 𝒮_n. This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in d^s, n+dd and ns+1 where d is the maximum degree of the input polynomials. When d,s are fixed, this is polynomial in n while when s is fixed and d ≃ n this yields an exponential speed-up with respect to the usual polynomial system solving algorithms.
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