Bivariate polynomial reduction and elimination ideal over finite fields

Given two polynomials a and b in $F_q[x,y]$ which have no non-trivial common divisors, we prove that a generator of the elimination ideal $〈a,b〉\cap F_q\left[x\right]$ can be computed in quasi-linear time. To achieve this, we propose a randomized algorithm of the Monte Carlo type which requires ${(de\log q)}^{1+o\left(1\right)}$ bit operations, where d and e bound the input degrees in x and in y respectively.

The same complexity estimate applies to the computation of the largest degree invariant factor of the Sylvester matrix associated with a and b (with respect to either x or y), and of the resultant of a and b if they are sufficiently generic, in particular such that the Sylvester matrix has a unique non-trivial invariant factor.

Our approach is to exploit reductions to problems of minimal polynomials in quotient algebras of the form $F_q[x,y]/〈a,b〉$. By proposing a new method based on structured polynomial matrix division for computing with the elements of the quotient, we succeed in improving the best-known complexity bounds.