Suppose you would like to count the exact number of common roots of n polynomials in n variables. Classical algebraic geometry only gives loose upper bounds on this number. More recent methods, e.g, mixed volumes, give sharper upper bounds but can still overcount sometimes. One could, of course, resort to Grobner bases but this route is still too slow for many practical applications, e.g., inverse kinematics. So what to do?
We propose a simple method, with great practical potential, based on the sparse resultant. Through the geometry of toric varieties we also obtain a new point of view on the technique of univariate reduction for solving polynomial systems. In particular, we obtain a toric variety generalization (the toric GCP) of a geometric projection operator proposed by Canny (the generalized characteristic polynomial). Canny's result gave one of the first known ``polynomial-time'' algorithm for solving polynomial systems. Our new toric GCP improves the previous complexity bounds to take the monomial term structure of a given polynomial system into account.
(No prior knowledge of toric varieties or sparse resultants is assumed.)