  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 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.

Combinatorics;Computational Geometry;Algebraic Geometry;Metric Geometry