# A numerical toolkit for multiprojective varieties

## TL;DR

This paper develops a numerical toolkit for analyzing multiprojective varieties using witness collections, enabling efficient numerical irreducible decomposition with practical examples.

## Contribution

It introduces a new toolkit for numerical manipulation of multiprojective varieties and presents an algorithm for their numerical irreducible decomposition.

## Key findings

- Toolkit effectively manipulates witness collections.
- Algorithm successfully performs numerical irreducible decomposition.
- Examples demonstrate practical applicability.

## Abstract

A numerical description of an algebraic subvariety of projective space is given by a general linear section, called a witness set. For a subvariety of a product of projective spaces (a multiprojective variety), the corresponding numerical description is given by a witness collection, whose structure is more involved. We build on recent work to develop a toolkit for the numerical manipulation of multiprojective varieties that operates on witness collections, and use this toolkit in an algorithm for numerical irreducible decomposition of multiprojective varieties. The toolkit and decomposition algorithm are illustrated throughout in a series of examples.