Algebraic boundary of matrices of nonnegative rank at most three (Q307808)

The nonnegative rank of a matrix \(M\in\mathbb R_{\geq 0}^{m\times n}\) is the smallest \(r\in\mathbb N\) such that there exist matrices \(A\in \mathbb R_{\geq 0}^{m\times r}\) and \(B\in \mathbb R_{\geq 0}^{r\times n}\) with \(M=AB\). Matrices of nonnegative rank at most \(r\) form a semialgebraic set, which is denoted by \(\mathcal M_{m\times n}^r\). The present paper studies the Zariski closure of the boundary of \(\mathcal M_{m\times n}^3\).NEWLINENEWLINEThe Zariski closure of the boundary of the set of matrices of nonnegative rank at most 3 is reducible. The main theorem in this paper (previously conjectured in [\textit{K. Kubjas} et al., Ann. Stat. 43, No. 1, 422--461 (2015; Zbl 1308.62035), Conjecture 6.4]), describes all boundary components of the set of \(m\times n\)-matrices of nonnegative rank 3. The authors give a minimal generating set for the ideal of each irreducible component. This generating set is a Gröbner basis with respect to the graded reverse lexicographic order.