Trace test

TL;DR

This paper discusses the trace test in numerical algebraic geometry, extending it to subvarieties of product spaces with a practical dimension reduction approach for efficient verification.

Contribution

It introduces a new method for applying the trace test to subvarieties of product spaces using multihomogeneous witness sets and dimension reduction techniques.

Findings

Effective trace test for subvarieties in product spaces

Dimension reduction simplifies the trace test process

Practical approach for verifying witness sets in affine space

Abstract

The trace test in numerical algebraic geometry verifies the completeness of a witness set of an irreducible variety in affine or projective space. We give a brief derivation of the trace test and then consider it for subvarieties of products of projective spaces using multihomogeneous witness sets. We show how a dimension reduction leads to a practical trace test in this case involving a curve in a low-dimensional affine space.