MIXED VOLUME COMPUTATION VIA LINEAR PROGRAMMING

Abstract

A renewed algorithm is presented to calculate the mixed volume of the support ${\cal A}=({\cal A}_1,\dots,{\cal A}_n)$ of a polynomial system $P({\bf x})=(p_1({\bf x}),\dots, p_n({\bf x}))$ in $\Bbb C^n$. The key ingredient is a specially tailored application of LP feasibility tests, which allows us to calculate the mixed cells, their volumes constituting the mixed volume, in a mixed subdivision of ${\cal A}$ more efficiently. The problem of finding mixed cells plays a crucial role in polyhedral homotopy methods for finding all isolated zeros of $P({\bf x})$. Our new algorithm advances the speed of mixed volume computation by a considerable margin, illustrated by numerical examples.