Elimination ideal and bivariate resultant over finite fields

TL;DR

This paper introduces a randomized algorithm for efficiently computing the largest invariant factor of the Sylvester matrix of two bivariate polynomials over finite fields, improving complexity bounds for elimination ideals and resultants.

Contribution

It presents a novel randomized method based on structured polynomial matrix division that enhances the efficiency of computing elimination ideals and resultants over finite fields.

Findings

Algorithm has complexity $O((de)^{1+psilon} ext{log}(q)^{1+o(1)})$

Efficient computation of elimination ideals when the system has no roots at infinity

Improved bounds for the computation of resultants of generic polynomials

Abstract

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.