Bezout number calculations for multi-homogeneous polynomial systems

Abstract

The Bezout number of a system of multi-homogeneous polynomial equations is the largest number of nonsingular solutions such a system can have, and it is also the number of solution paths used to compute all geometrically isolated solutions of the system using multi-homogeneous polynomial continuation. Any polynomial system can be homogenized in a variety of ways, typically yielding different Bezout numbers. This paper presents an efficient algorithm for computing Bezout numbers and describes a procedure for finding the minimal Bezout number over all possible homogenizations.