Combinatorial Resultants in the Algebraic Rigidity Matroid

TL;DR

This paper introduces a novel algorithm for computing circuit polynomials in the algebraic rigidity matroid of 2D points, leveraging combinatorial resultants to improve efficiency over traditional Groebner Basis methods.

Contribution

It develops a new combinatorial operation called resultants, enabling faster algebraic elimination in rigidity matroids, with a construction tree approach from K4 graphs.

Findings

Algorithm computes circuit polynomials significantly faster than Groebner Basis methods.

Introduces combinatorial resultants capturing properties of Sylvester resultants in rigidity matroids.

Demonstrates practical efficiency with a Mathematica implementation that outperforms traditional algebraic methods.

Abstract

Motivated by a rigidity-theoretic perspective on the Localization Problem in 2D, we develop an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley-Menger ideal for $n$ points in 2D. We introduce combinatorial resultants, a new operation on graphs that captures properties of the Sylvester resultant of two polynomials in the algebraic rigidity matroid. We show that every rigidity circuit has a construction tree from $K_4$ graphs based on this operation. Our algorithm performs an algebraic elimination guided by the construction tree, and uses classical resultants, factorization and ideal membership. To demonstrate its effectiveness, we implemented our algorithm in Mathematica: it took less than 15 seconds on an example where a Groebner Basis calculation took 5 days and 6 hrs.