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Preprints, Working Papers, ... Year : 2024

Bivariate polynomial reduction and elimination ideal over finite fields

Abstract

Given two polynomials a and b in Fq[x,y] which have no non-trivial common divisors, we prove that a generator of the elimination ideal ⟨a, b⟩ ∩ Fq [x] 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(1) 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 Fq [x, y]/⟨a, b⟩. By proposing a new method based on structured polyno- mial matrix division for computing with the elements of the quotient, we succeed in improving the best-known complexity bounds.
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Dates and versions

hal-04542799 , version 1 (11-04-2024)

Identifiers

  • HAL Id : hal-04542799 , version 1

Cite

Gilles Villard. Bivariate polynomial reduction and elimination ideal over finite fields. 2024. ⟨hal-04542799⟩
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