Partial Results on the Degree of the Algebraic Boundary in Rank-One Tensor Completion

Kahle, Kubjas, Kummer and Rosen showed that for a set E of observed entries of a d_1 × ⋯ × d_n tensor with |E| = Σ_j (d_j − 1) that meets every maximal slice and has a full-dimensional completable region (we call such E admissible), the algebraic boundary of the set of partial tensors that are restrictions of joint distributions of independent random variables consists of coordinate hyperplanes and one irreducible hypersurface H. They asked for deg H as a function of n, d_1, …, d_n and E. We give a partial answer. For the corner patterns, the cells that differ from a fixed cell in exactly one coordinate, H is defined by an explicit irreducible discriminant and deg H = n(n − 1) in every format. In general deg H ≤ m!/∏_j (d_j − 1)! with m = |E|, with equality for some patterns. We call E saturated if the maximal minors of its incidence matrix are coprime; this is weaker than unimodularity of that matrix. If E is saturated, the complex completions of a generic partial tensor are the W roots of one Laurent polynomial equation Φ_x(τ) = 1, and we prove q·deg H = W* + deg Den, where W* ≤ W counts the critical values of Φ_x that are not identically zero, q ≥ 1, and Den is a constant multiple of the q-th power of the lowest-degree form of the equation of H. For W = 2 this gives an explicit formula; for matrices deg H = 2 + 2p, where p is the number of observed entries that share their row and their column with other observed entries. For saturated patterns in binary formats we obtain deg H ≤ W + Σ_e μ_e with explicit tropical multiplicities. With exact computer certificates this determines deg H for all admissible patterns in the six formats 2 × 2, 3 × 3, 3 × 4, 2 × 2 × 2, 2 × 2 × 3 and 2 × 2 × 2 × 2. In a modular census of 2343 patterns, which is not a proof, all saturated patterns have q = 1 and W* = W, and the tropical bound is sharp; we do not prove this in general. For patterns that are not saturated we have only the general bound and one certified example. This is an unrefereed note.