Liftings of polynomial systems decreasing the mixed volume

TL;DR

This paper introduces a method to reduce the mixed volume of polynomial systems by lifting them to equivalent systems with more variables, improving the efficiency of homotopy continuation solvers.

Contribution

It presents a novel lifting strategy that decreases mixed volume for polynomial systems, enhancing solution efficiency in computational algebraic geometry.

Findings

Lifting can lower mixed volume in bivariate systems.

Lifting applies to certain dense polynomial systems.

Improves path-tracking efficiency in solvers.

Abstract

The BKK theorem states that the mixed volume of the Newton polytopes of a system of polynomial equations upper bounds the number of isolated torus solutions of the system. Homotopy continuation solvers make use of this fact to pick efficient start systems. For systems where the mixed volume bound is not attained, such methods are still tracking more paths than necessary. We propose a strategy of improvement by lifting a system to an equivalent system with a strictly lower mixed volume at the expense of more variables. We illustrate this idea providing lifting constructions for arbitrary bivariate systems and certain dense-enough systems.