Multiprojective witness sets and a trace test

TL;DR

This paper introduces multiprojective witness sets in numerical algebraic geometry, generalizing existing methods for positive-dimensional solution sets to handle multidegree information of multiprojective varieties.

Contribution

It extends the regeneration solving procedure, trace test, and numerical irreducible decomposition to the multiprojective setting, providing a new approach for complex polynomial systems.

Findings

Generalization of the trace test to multiprojective varieties

Development of multiprojective witness sets for multidegree encoding

Demonstration of the approach through examples

Abstract

In the field of numerical algebraic geometry, positive-dimensional solution sets of systems of polynomial equations are described by witness sets. In this paper, we define multiprojective witness sets which encode the multidegree information of an irreducible multiprojective variety. Our main results generalize the regeneration solving procedure, a trace test, and numerical irreducible decomposition to the multiprojective case. Examples are included to demonstrate this new approach.