Document Type
Conference Proceeding
Publication Date
6-1-2021
Publication Title
Leibniz International Proceedings in Informatics, LIPIcs
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.
Keywords
Cayley-Menger ideal, Circuit polynomial, Combinatorial resultant, Gröbner basis elimination, Inductive construction, Rigidity matroid
Volume
189
DOI
10.4230/LIPIcs.SoCG.2021.52
ISSN
18688969
Rights
© 2021 by the authors.
Recommended Citation
Malić, Goran and Streinu, Ileana, "Combinatorial resultants in the algebraic rigidity matroid" (2021). Computer Science: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/csc_facpubs/289
Comments
Peer reviewed accepted manuscript.