Multivariate power series multiplication

Abstract

We study the multiplication of multivariate power series. We show that over large enough fields, the bilinear complexity of the product modulo a monomial ideal M is bounded by the product of the regularity of M by the degree of M . In some special cases, such as partial degree truncation, this estimate carries over to total complexity. This leads to complexity improvements for some basic algorithms with algebraic numbers, and some polynomial system solving algorithms.

Fast arithmetic for triangular sets

“…For the question of power series multiplication, no quasi-linear algorithm was known until (29). We extend this result to the case of arbitrary T i ∈ K[X i ], the crucial question being how to avoid expanding the (polynomial) product AB before reducing it.…”

Multi-point evaluation in higher dimensions

“…For more general truncation patterns, Schost [Sch05] introduced an algorithm based on deformation techniques that uses evaluation and interpolation of the form described in this paper. At the time of writing [Sch05], no efficient algorithm was known for evaluation and interpolation; the present paper fills this gap and completes the results of [Sch05].…”

“…Let us first recall an algorithm of [Sch05] and show how our results enable us to improve it. Theorem 1 of [Sch05] gives an algorithm for multiplication in C[x]/m, that relies on the following operations:…”

“…Theorem 1 of [Sch05] gives an algorithm for multiplication in C[x]/m, that relies on the following operations:…”

“…The paper [Sch05] does not specify how to do the evaluation and interpolation (for lack of an efficient solution); using our results, it becomes possible to fill all the gaps in this algorithm. Applying Theorem 5, without doing any simplification, we obtain a cost of…”