Chordal Networks of Polynomial Ideals
Name
16m106995x.pdf
Description
Published version
Size
1.25 MB
Format
Adobe PDF
Checksum (MD5)
038c8908fe525af6ac6e3dc350961163
Author(s)
Cifuentes, Diego Fernando
Parrilo, Pablo A.
Date Issued
January 2017
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Citation
Cifuentes, Diego, and Pablo A. Parrilo. “Chordal Networks of Polynomial Ideals.” SIAM Journal on Applied Algebra and Geometry 1, no. 1 (January 2017): 73–110. © 2017 Society for Industrial and Applied Mathematics
Version
Final published version
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. Key words. chordal graphs, structured polynomials, chordal networks, triangular sets
MIT Department
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/16m106995x
