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- W4287674155 abstract "Let $mathbf{K}$ be a field and $phi$, $mathbf{f} = (f_1, ldots, f_s)$ in $mathbf{K}[x_1, dots, x_n]$ be multivariate polynomials (with $s < n$) invariant under the action of $mathcal{S}_n$, the group of permutations of ${1, dots, n}$. We consider the problem of computing the points at which $mathbf{f}$ vanish and the Jacobian matrix associated to $mathbf{f}, phi$ 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 $mathcal{S}_n$. This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in $d^s$, ${{n+d}choose{d}}$ 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." @default.
- W4287674155 created "2022-07-25" @default.
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- W4287674155 date "2020-09-01" @default.
- W4287674155 modified "2023-10-13" @default.
- W4287674155 title "Computing critical points for invariant algebraic systems" @default.
- W4287674155 hasPublicationYear "2020" @default.
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