Combinatorial Resultants in the Algebraic Rigidity Matroid [article]

Goran Malić, Ileana Streinu
2021 arXiv   pre-print
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
more » ... n. 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.
arXiv:2103.08432v1 fatcat:psdqebn445bwfdsx3tqf7ovute