On the complexity of computing with zero-dimensional triangular sets

Abstract

International audienceWe study the complexity of some fundamental operations for triangular sets in dimension zero. Using Las Vegas algorithms, we prove that one can perform such operations as change of order, equiprojectable decomposition, or quasi-inverse computation with a cost that is essentially that of modular composition. Over an abstract field, this leads to a subquadratic cost (with respect to the degree of the underlying algebraic set). Over a finite field, in a boolean \RAM\ model, we obtain a quasi-linear running time using Kedlaya and Umansʼ algorithm for modular composition. Conversely, we also show how to reduce the problem of modular composition to change of order for triangular sets, so that all these problems are essentially equivalent. Our algorithms are implemented in Maple; we present some experimental results

Algorithms for the universal decomposition algebra

“…Starting from the divided differences, the RUR algorithm [35] or the FGLM algorithm [20] take time O(δ 3 n ); a recent improvement of the latter [21] could reduce the exponent using sparse linear algebra techniques. Some other algorithms are specifically designed to take as input a triangular set (such as the divided differences) and convert it to a univariate representation, such as [10] or [32]; the latter performs the conversion for any m in subquadratic time…”

“…Following [18,32], our main idea is to introduce mixed representations, that allow one to convert from triangular to bivariate representations, and back, one variable at a time.…”

“…. , R T ,m−1 ; in the introduction, we only defined univariate representations for the algebras Ai, but the definition carries over unchanged to this slightly more general context [23,32]. For i m − 1, Pi has the form Pi = (Qi, Si,1, .…”

“…Two operations will be needed to convert between the various induced representations: lift-up and push-down [18,32]. For i m − 2 and i + 1 j m, we call lift-up the change of basis up i,j := Φi+1,j • Φ −1 i,j .…”

“…downi) coefficientwise. We do not discuss here how to implement them in general (see [32]); we will give a better solution in the case of divided differences below. For the moment, we simply record the following straightforward result.…”