Work

Computing Circuit Polynomials in the Algebraic Rigidity Matroid

Year: 2023

Type: preprint

Abstract: We present an algorithm for computing circuit polynomials in the algebraic rigidity matroid $\mathcal{A}(\text{CM}_n)$ 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-$K_4$ generators of the Cayley-Menger ideal and simple variations on our main algorithm.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2304.12435

Authors: Malic, Goran, Streinu, Ileana

Institution: Smith College

Language: English

Topic: Polynomial and algebraic computation

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: AF:Medium:RUI:Algorithmic Problems in Kinematic Distance Geometry, AF:Medium:Collaborative:RUI:Structure in Motion:Algorithms for Kinematic Design

OpenAlex ID: W4367060540

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