On the complexity of computing with zero-dimensional triangular sets

TL;DR

This paper analyzes the computational complexity of operations on zero-dimensional triangular sets, providing algorithms with near-optimal performance and establishing equivalences between key problems in algebraic computation.

Contribution

It introduces new algorithms for operations on triangular sets with subquadratic and quasi-linear complexity, and shows the equivalence of modular composition and change of order problems.

Findings

Algorithms achieve subquadratic cost over abstract fields.

Quasi-linear time algorithms are obtained over finite fields using Kedlaya and Umans' method.

Experimental results demonstrate practical efficiency of the algorithms.

Abstract

We 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 Umans' 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.