Chordal networks of polynomial ideals

04/09/2016
by   Diego Cifuentes, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
02/06/2018

On the chordality of polynomial sets in triangular decomposition in top-down style

In this paper the chordal graph structures of polynomial sets appearing ...
research
11/25/2018

Chordal Graphs in Triangular Decomposition in Top-Down Style

In this paper, we first prove that when the associated graph of a polyno...
research
08/01/2022

Graphical Representations for Algebraic Constraints of Linear Structural Equations Models

The observational characteristics of a linear structural equation model ...
research
05/10/2016

Need Polynomial Systems be Doubly-exponential?

Polynomial Systems, or at least their algorithms, have the reputation of...
research
02/06/2023

Short proofs of ideal membership

A cofactor representation of an ideal element, that is, a representation...
research
01/25/2022

Computing the logarithmic capacity of compact sets having (infinitely) many components with the Charge Simulation Method

We apply the Charge Simulation Method (CSM) in order to compute the loga...

Please sign up or login with your details

Forgot password? Click here to reset