Polynomial Solvers for Saturated Ideals

In this paper we present a new method for creating polynomial solvers for problems where a (possibly infinite) subset of the solutions are undesirable or uninteresting. These solutions typically arise from simplifications made during modeling, but can also come from degeneracies which are inherent to the geometry of the original problem. The proposed approach extends the standard action matrix method to saturated ideals. This allows us to add constraints that some polynomials should be non-zero on the solutions. This does not only offer the possibility of improved performance by removing superfluous solutions, but makes a larger class of problems tractable. Previously, problems with infinitely many solutions could not be solved directly using the action matrix method as it requires a zero-dimensional ideal. In contrast we only require that after removing the unwanted solutions only finitely many remain. We evaluate our method on three applications, optimal triangulation, time-of-arrival self-calibration and optimal vanishing point estimation.