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A graph profile records all possible densities of a fixed finite set of graphs. Profiles can be extremely complicated; for instance the full profile of any triple of connected graphs is not known, and little is known about hypergraph profiles. We introduce the tropicalization of graph and hypergraph profiles. Tropicalization is a well-studied operation in algebraic geometry, which replaces a variety (the set of real or complex solutions to a finite set of algebraic equations) with its "combinatorial shadow". We prove that the tropicalization of a graph profile is a closed convex cone, which still captures interesting combinatorial information. We explicitly compute these tropicalizations for arbitrary sets of complete and star hypergraphs. We show they are rational polyhedral cones even though the corresponding profiles are not even known to be semialgebraic in some of these cases. We then use tropicalization to prove strong restrictions on the power of the sums of squares method, equivalently Cauchy-Schwarz calculus, to test (which is weaker than certification) the validity of graph density inequalities. In particular, we show that sums of squares cannot test simple binomial graph density inequalities, or even their approximations. Small concrete examples of such inequalities are presented, and include the famous Blakley-Roy inequalities for paths of odd length. As a consequence, these simple inequalities cannot be written as a rational sum of squares of graph densities. [ABSTRACT FROM AUTHOR]

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]

We study symmetric non-negative forms and their relationship with symmetric sums of squares. For a fixed number of variables n and degree 2d, symmetric non-negative forms and symmetric sums of squares form closed, convex cones in the vector space of n-variate symmetric forms of degree 2d. Using representation theory of the symmetric group we characterize both cones in a uniform way. Further, we investigate the asymptotic behavior when the degree 2d is fixed and the number of variables n grows. Here, we show that, in sharp contrast to the general case, the difference between symmetric non-negative forms and sums of squares does not grow arbitrarily large for any fixed degree 2d. We consider the case of symmetric quartic forms in more detail and give a complete characterization of quartic symmetric sums of squares. Furthermore, we show that in degree 4 the cones of non-negative symmetric forms and symmetric sums of squares approach the same limit, thus these two cones asymptotically become closer as the number of variables grows. We conjecture that this is true in arbitrary degree 2d. [ABSTRACT FROM AUTHOR]

Establishing inequalities among graph densities is a central pursuit in extremal combinatorics. A standard tool to certify the nonnegativity of a graph density expression is to write it as a sum of squares. In this paper, we identify a simple condition under which a graph density expression cannot be a sum of squares. Using this result, we prove that the Blakley-Roy inequality does not have a sum of squares certificate when the path length is odd. We also show that the same Blakley-Roy inequalities cannot be certified by sums of squares using a multiplier of the form one plus a sum of squares. These results answer two questions raised by Lovász. Our main tool is used again to show that the smallest open case of Sidorenko's conjectured inequality cannot be certified by a sum of squares. Finally, we show that our setup is equivalent to existing frameworks by Razborov and Lovász-Szegedy, and thus our results hold in these settings too. [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 article examines the connection between varieties of minimal degree and sums of squares. Topics covered include the relation of nonnegativity with sums of squares, generalization of Hilbert's 1888 classification of nonnegative firms, connection between real and complex algebraic geometry and list of varieties for which nonnegative quadratic forms are sums of squares.

We study dimensions of the faces of the cone of nonnegative polynomials and the cone of sums of squares; we show that there are dimensional differences between corresponding faces of these cones. These dimensional gaps occur in all cases where there exist nonnegative polynomials that are not sums of squares. The gaps occur generically, they are not the product of selecting special faces of the cones. For ternary forms and quaternary quartics, we characterize when these differences are observed. Moreover, we provide an explicit description for these differences in the two smallest cases, in which the cone of nonnegative polynomials and the cone of sums of squares are different. Our results follow from more general results concerning the relationship between the degree 2dcomponent of the second symbolic power of the vanishing ideal of points in projective space and the square of the degree d component of the vanishing ideal. [ABSTRACT FROM AUTHOR]