Reduction of bivariate polynomials from convex-dense to dense, with application to factorizations

Abstract

International audienceIn this article we present a new algorithm for reducing the usual sparse bivariate factorization problems to the dense case. This reduction simply consists in computing an invertible monomial transformation that produces a polynomial with a dense size of the same order of magnitude as the size of the integral convex hull of the support of the input polynomial. This approach turns out to be very efficient in practice, as demonstrated with our implementation

On the complexity of the Lickteig–Roy subresultant algorithm

“…In addition, the underlined expressions may be discarded when B is a eld. Let us mention that complexities in terms of convex hulls of the supports of A and B may further be derived thanks to the algorithm in [10]. Previously such deterministic costs for bivariate gcds were involving hypotheses on the cardinality of K in order to use fast evaluation/interpolation strategies, as in [24,Chapter 10].…”

Multivariate sparse interpolation using randomized Kronecker substitutions

“…There is recent work on implementations of sparse interpolation Javadi and Monagan [2010], van der Hoeven and Lecerf [2014] and applications to GCD and factorization Monagan, 2009, Berthomieu andLecerf, 2012]. The Kronecker substitution in particular has been applied to integer and polynomial multiplication [Schönhage, 1982, Harvey, 2009.…”

Bounded-degree factors of lacunary multivariate polynomials

“…To really have an algorithm to compute bounded-degree factors of bivariate polynomials, one can branch any bivariate factorization algorithm. In order to get the best complexity bounds, one can preprocess the output polynomials before their factorization with the techniques of Berthomieu and Lecerf [4]. This allows to compute the irreducible factorization of a polynomial in time polynomial in the convex size rather than the degree of the polynomial.…”

“…The call to Bipartition takes polynomial time. Now, using the techniques of Berthomieu and Lecerf [4], one can compute the gcd of S in time polynomial in the convex size of the elements of S. This convex size is bounded by O(d 2 X d 2 Y k 4 ) according to Lemma 4.10. It remains to prove that one can give similar algorithms for the factors of f that have two non-parallel edges in the upper hull of the Newton polygon, or two vertical edges.…”