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We bound the Pythagoras number of a real projective subvariety: the smallest positive integer r such that every sum of squares of linear forms in its homogeneous coordinate ring is a sum of at most r squares. Enhancing existing methods, we exhibit three distinct upper bounds involving known invariants. In contrast, our lower bound depends on a new invariant of a projective subvariety called the quadratic persistence. Defined by projecting away from points, this numerical invariant is closely related to the linear syzygies of the variety. In addition, we classify the projective subvarieties of maximal and almost-maximal quadratic persistence, and determine their Pythagoras numbers. [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]

The cone of sums of squares is one of the central objects in convex algebraic geometry. Its defining linear inequalities correspond to the extreme rays of the dual convex cone. This dual cone is a spectrahedron, which can be explicitly realized as a section of the cone of positive semidefinite matrices with the linear subspace of Hankel (or middle catalecticant) matrices. In this paper we initiate a systematic study of the extreme rays of Hankel spectrahedra for ternary forms. We show that the Zariski closure of the union of extreme rays is the variety of all Hankel matrices of corank at least 4, an irreducible variety of codimension 10 and we determine its degree. We explicitly construct an extreme ray of maximal rank using the Cayley–Bacharach Theorem for plane curves. We apply our results to the study of the algebraic boundary of the cone of sums of squares. Its irreducible components are dual to varieties of Gorenstein ideals with certain Hilbert functions. We determine these Hilbert functions for some cases of small degree. We also observe surprising gaps in the ranks of Hankel matrices of the extreme rays. [ABSTRACT FROM AUTHOR]

Let X ⊆ P m be a totally real, non-degenerate, projective variety and let Γ ⊆ X ( R ) be a generic set of points. Let P be the cone of nonnegative quadratic forms on X and let Σ be the cone of sums of squares of linear forms. We examine the dimensions of the faces P ( Γ ) and Σ ( Γ ) consisting of forms in P and Σ, which vanish on Γ. As the cardinality of the set Γ varies in 1 , 2 , … , codim ( X ) , the difference between the dimensions of P ( Γ ) and Σ ( Γ ) defines a numerical invariant of X , which we call the gap vector of X . Our main result is a formula relating the components of the gap vector of X and the quadratic deficiencies of X and its generic projections. Using it, we prove that gap vectors are weakly increasing, obtain upper bounds for their rate of growth and prove that these upper bounds are eventually achieved for all varieties. Moreover, we give a characterization of the varieties with the simplest gap vectors: we prove that the gap vector vanishes identically precisely for varieties of minimal degree, giving another proof that P ≠ Σ when X is not a variety of minimal degree [4] . We also characterize the varieties whose gap vector equals ( 0 , … , 0 , 1 ) . [ABSTRACT FROM AUTHOR]