Combinatorial Resultants in the Algebraic Rigidity Matroid

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 CMn 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 K4 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 Gröbner Basis calculation took 5 days and 6 hrs. 2012 ACM Subject Classification General and reference → Performance; General and reference → Experimentation; Theory of computation → Computational geometry; Mathematics of computing → Matroids and greedoids; Mathematics of computing → Mathematical software performance; Computing methodologies → Combinatorial algorithms; Computing methodologies → Algebraic algorithms