Chordal networks of polynomial ideals

TL;DR

This paper introduces chordal networks, a new structured representation for polynomial ideals that exploits graph sparsity to enable efficient computation of algebraic properties and outperforms existing methods in practical applications.

Contribution

The paper presents chordal networks as a novel, compact representation for polynomial ideals that preserves graphical structure and improves computational efficiency.

Findings

Chordal networks can represent many polynomial ideals with size linear in variables.

They enable efficient computation of variety properties like dimension and cardinality.

Algorithms based on chordal networks outperform existing techniques in algebraic statistics and vector addition systems.

Abstract

We introduce a novel representation of structured polynomial ideals, which we refer to as chordal networks. The sparsity structure of a polynomial system is often described by a graph that captures the interactions among the variables. Chordal networks provide a computationally convenient decomposition into simpler (triangular) polynomial sets, while preserving the underlying graphical structure. We show that many interesting families of polynomial ideals admit compact chordal network representations (of size linear in the number of variables), even though the number of components is exponentially large. Chordal networks can be computed for arbitrary polynomial systems using a refinement of the chordal elimination algorithm from [Cifuentes-Parrilo-2016]. Furthermore, they can be effectively used to obtain several properties of the variety, such as its dimension, cardinality, and equidimensional components, as well as an efficient probabilistic test for radical ideal membership. We apply our methods to examples from algebraic statistics and vector addition systems; for these instances, algorithms based on chordal networks outperform existing techniques by orders of magnitude.