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Communication Dans Un Congrès Année : 2023

Elimination ideal and bivariate resultant over finite fields

Résumé

A new algorithm is presented for computing the largest degree invariant factor of the Sylvester matrix (with respect either to $x$ or $y$) associated to two polynomials $a$ and $b$ in $\mathbb F_q[x,y]$ which have no non-trivial common divisors. The algorithm is randomized of the Monte Carlo type and requires $O((de)^{1+\epsilon}\log(q) ^{1+o(1)})$ bit operations, where $d$ an $e$ respectively bound the input degrees in $x$ and in $y$. It follows that the same complexity estimate is valid for computing: a generator of the elimination ideal $\langle a,b \rangle \cap \mathbb F_q[x]$ (or $\mathbb F_q[y]$), as soon as the polynomial system $a=b=0$ has not roots at infinity; the resultant of $a$ and $b$ when they are sufficiently generic, especially so that the Sylvester matrix has a unique non-trivial invariant factor. Our approach is to use the reduction of the problem to a problem of minimal polynomial in the quotient algebra $\mathbb F_q[x,y]/\langle a,b \rangle$. By proposing a new method based on structured polynomial matrix division for computing with the elements in the quotient, we manage to improve the best known complexity bounds.
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Dates et versions

hal-03999414 , version 1 (21-02-2023)

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Paternité

Identifiants

Citer

Gilles Villard. Elimination ideal and bivariate resultant over finite fields. ISSAC 2023: International Symposium on Symbolic and Algebraic Computation 2023, Jul 2023, Tromsø Norway, Norway. pp.526-534, ⟨10.1145/3597066.3597100⟩. ⟨hal-03999414⟩
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