Balancing the lifting values to improve the numerical stability of polyhedral homotopy continuation methods

Abstract

Polyhedral homotopy continuation methods exploit the sparsity of polynomial systems so that the number of solution curves to reach all isolated solutions is optimal for generic systems. The numerical stability of tracing solution curves of polyhedral homotopies is mainly determined by the height of the powers of the continuation parameter. To reduce this height, we propose a procedure that operates as an intermediate stage between the mixed-volume computation and the tracing of solution curves. This procedure computes new lifting values of the support of a polynomial system. These values preserve the structure of the mixed-cell configuration obtained from the mixed-volume computation and produce better-balanced powers of the continuation parameter in the polyhedral homotopies.