Algebraic boundary of matrices of nonnegative rank at most three

TL;DR

This paper characterizes the algebraic boundary of matrices with nonnegative rank at most three, providing minimal generating sets and Gr"obner bases for each component, thus resolving a conjecture in algebraic geometry.

Contribution

It identifies the irreducible components of the boundary and constructs explicit Gr"obner bases, advancing understanding of nonnegative matrix rank boundaries.

Findings

The boundary is reducible with multiple components.

Explicit minimal generating sets are provided for each component.

A Gr"obner basis is established for the ideal of each irreducible component.

Abstract

The Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Grobner basis with respect to the graded reverse lexicographic order. This solves a conjecture by Robeva, Sturmfels and the last author.