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Journal Articles Journal of Symbolic Computation Year : 2023

Computing critical points for invariant algebraic systems

Jean-Charles Faugère
Mohab Safey El Din
Éric Schost
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Thi Xuan Vu


Let $\KK$ be a field and $\phi$, $\f = (f_1, \ldots, f_s)$ in $\KK[x_1, \dots, x_n]$ be multivariate polynomials (with $s < n$) invariant under the action of $\Sc_n$, the group of permutations of $\{1, \dots, n\}$. We consider the problem of computing the points at which $\f$ vanish and the Jacobian matrix associated to $\f, \phi$ 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 $\Sc_n$. This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in $d^s$, ${{n+d}\choose{d}}$ and $\binom{n}{s+1}$ 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 $d \simeq n$ this yields an exponential speed-up with respect to the usual polynomial system solving algorithms.
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hal-02927636 , version 1 (01-09-2020)



Jean-Charles Faugère, George Labahn, Mohab Safey El Din, Éric Schost, Thi Xuan Vu. Computing critical points for invariant algebraic systems. Journal of Symbolic Computation, 2023, 116, pp.365-399. ⟨10.1016/j.jsc.2022.10.002⟩. ⟨hal-02927636⟩
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