Computing critical points for invariant algebraic systems

09/02/2020
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by   Jean-Charles Faugรจre, et al.
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Let ๐Š be a field and ฯ•, ๐Ÿ = (f_1, โ€ฆ, f_s) in ๐Š[x_1, โ€ฆ, x_n] be multivariate polynomials (with s < n) invariant under the action of ๐’ฎ_n, the group of permutations of {1, โ€ฆ, n}. We consider the problem of computing the points at which ๐Ÿ vanish and the Jacobian matrix associated to ๐Ÿ, ฯ• 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 ๐’ฎ_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 โ‰ƒ n this yields an exponential speed-up with respect to the usual polynomial system solving algorithms.

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