2021/03/15 by Goran Malić, Malić, Goran, Ileana Streinu +1 · 1 citation
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Metric Geometry (math.MG) #cs.CG #math.AG #math.CO #math.MG
paper · pdf · doi:10.48550/arxiv.2103.08432
32 pages, 9 figures. This is the full paper accompanying the extended abstract of the same title (to appear in SoCG 2021)
arxiv created 2021/03/15 · arxiv updated 2021/03/17
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 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 Groebner Basis calculation took 5 days and 6 hrs.