Computing Circuit Polynomials in the Algebraic Rigidity Matroid
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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-K_4 generators of the Cayley-Menger ideal and simple variations on our main algorithm.
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