A Direttissimo Algorithm for Equidimensional Decomposition

TL;DR

This paper introduces a recursive algorithm that decomposes algebraic sets into equidimensional components without relying on projections or genericity assumptions, improving efficiency and structure preservation.

Contribution

It presents a novel algorithm combining triangular sets and Gröbner bases for equidimensional decomposition that avoids projections and genericity assumptions.

Findings

Efficient decomposition of algebraic sets demonstrated.

Algorithm outperforms existing methods in practical experiments.

Produces fine decompositions preserving system structure.

Abstract

We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the theory of triangular sets, a.k.a. regular chains, with Gr\"obner bases to encode and work with locally closed algebraic sets. Equipped with this, our algorithm avoids projections of the algebraic sets that are decomposed and certain genericity assumptions frequently made when decomposing polynomial systems, such as assumptions about Noether position. This makes it produce fine decompositions on more structured systems where ensuring genericity assumptions often destroys the structure of the system at hand. Practical experiments demonstrate its efficiency compared to state-of-the-art implementations.