Numerical Solution of Bivariate and Polyanalytic Polynomial Systems

Abstract

Finding the real solutions of a bivariate polynomial system is a central problem in robotics, computer modeling and graphics, computational geometry, and numerical optimization. We propose an efficient and numerically robust algorithm for solving bivariate and polyanalytic polynomial systems using a single generalized eigenvalue decomposition. In contrast to existing eigen-based solvers, the proposed algorithm does not depend on Gröbner bases or normal sets, nor does it require computing eigenvectors or solving additional eigenproblems to recover the solution. The method transforms bivariate systems into polyanalytic systems and then uses resultants in a novel way to project the variables onto the real plane associated with the two variables. Solutions are returned counting multiplicity and their accuracy is maximized by means of numerical balancing and Newton--Raphson refinement. Numerical experiments show that the proposed algorithm consistently recovers a higher percentage of solutions and is at the same time significantly faster and more accurate than competing double precision solvers.