Computing Circuit Polynomials in the Algebraic Rigidity Matroid

We present an algorithm for computing circuit polynomials in the algebraic rigidity matroid associated to the Cayley–Menger ideal for 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 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 hours. Additional speed-ups are obtained using non- generators of the Cayley–Menger ideal and simple variations on our main algorithm.