Computing critical points for invariant algebraic systems
share
Let ๐ be a field and ฯ, ๐ = (f_1, โฆ, f_s) in ๐[x_1, โฆ, x_n] be multivariate polynomials (with s < n) invariant under the action of ๐ฎ_n, the group of permutations of {1, โฆ, n}. We consider the problem of computing the points at which ๐ vanish and the Jacobian matrix associated to ๐, 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 ๐ฎ_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 n this yields an exponential speed-up with respect to the usual polynomial system solving algorithms.
READ FULL TEXT
