Computing Circuit Polynomials in the Algebraic Rigidity Matroid

Authors

Goran Malić, Smith College
Ileana Streinu, Smith CollegeFollow

Document Type

Article

Publication Date

4-2023

Publication Title

SIAM Journal on Applied Algebra and Geometry

Abstract

We present an algorithm for computing circuit polynomials in the algebraic rigidity matroid A(CM_n) associated to the Cayley-Menger ideal CM_n for n points in 2D. It relies on combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in this ideal. We show that every rigidity circuit has a construction tree from K₄ graphs based on this operation. Our algorithm performs an algebraic elimination guided by such a construction tree, and uses classical resultants, factorization and ideal membership. To highlight its effectiveness, we implemented the algorithm in Mathematica: it took less than 15 seconds on an example where a Gröbner Basis calculation took 5 days and 6 hrs. Additional speed-ups are obtained using non-K₄ generators of the Cayley-Menger ideal and simple variations on our main algorithm.

Keywords

Cayley–Menger ideal, rigidity matroid, circuit polynomial, combinatorial resultant, inductive construction, Gröbner basis elimination

Volume

7

Issue

2

DOI

https://doi.org/10.1137/21M143798

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Rights

©2023 SIAM

Comments

Archived as published.

Recommended Citation

Malić, Goran and Streinu, Ileana, "Computing Circuit Polynomials in the Algebraic Rigidity Matroid" (2023). Computer Science: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/csc_facpubs/400