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We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M , we keep the positions of specified and unspecified entries fixed, and study the minimal completion rank. If the entries of the matrix are complex and the known entries are chosen randomly according to a continuous distribution, then for a fixed pattern of locations of specified and unspecified entries, there is a unique minimum completion rank which occurs with probability one. We call this rank the generic completion rank. Over the real numbers there can be multiple ranks that occur with positive probability; we call them typical completion ranks. We introduce these notions formally, and provide a number of inequalities and exact results on typical and generic ranks for different families of patterns of known and unknown entries. [ABSTRACT FROM AUTHOR]

Given a real projective curve with homogeneous coordinate ring R and a nonnegative homogeneous element f ∈ R , we bound the degree of a nonzero homogeneous sum of squares g ∈ R such that the product fg is again a sum of squares. Better yet, our degree bounds only depend on geometric invariants of the curve and we show that there exist smooth curves and nonnegative elements for which our bounds are sharp. We deduce the existence of a multiplier g from a new Bertini Theorem in convex algebraic geometry and prove sharpness by deforming rational Harnack curves on toric surfaces. Our techniques also yield similar bounds for multipliers on surfaces of minimal degree, generalizing Hilbert's work on ternary forms. [ABSTRACT FROM AUTHOR]

We study the real rank of points with respect to a real variety X . This is a generalization of various tensor ranks, where X is in a specific family of real varieties like Veronese or Segre varieties. The maximal real rank can be bounded in terms of the codimension of X only. We show constructively that there exist varieties X for which this bound is tight. The same varieties provide examples where a previous bound of Blekherman–Teitler on the maximal X -rank is tight. We also give examples of varieties X for which the gap between maximal complex and the maximal real rank is arbitrarily large. To facilitate our constructions we prove a conjecture of Reznick on the maximal real symmetric rank of symmetric bivariate tensors. Finally we study the geometry of the set of points of maximal real rank in the case of real plane curves. [ABSTRACT FROM AUTHOR]