arXiv subjects

Grigoriy Blekherman

Publications and source records attributed to Grigoriy Blekherman.

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arXiv2014

Gap vectors of real projective varieties

Let $X\subseteq \mathbb{P}^m$ be a totally real, non-degenerate, projective variety and let $Γ\subseteq X(\mathbb{R})$ 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 $1,\dots,\rm{codim}(X)$, 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. In this article we begin a systematic study of its fundamental properties. 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. The quadratic deficiency is a fundamental numerical invariant of projective varieties introduced by F. L. Zak. The relationship between quadratic deficiency and gap vectors allows us to effectively compute the gap vectors of concrete varieties as well as to prove several general properties. 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, and characterize the varieties whose gap vector equals $(0,\dots, 0,1)$. In particular, our results give a new proof of the theorem of Blekherman, Smith and Velasco saying that there are nonnegative quadratic forms which are not sums of squares on every variety, which is not of minimal degree. Finally, we determine the gap vector of all Veronese embeddings of $\mathbb{P}^2$, generalizing work of the first three authors.

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arXiv2014

Sums of Squares on the Hypercube

Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we are interested in rational function representations of small degree. We derive a general upper bound in terms of the Hilbert function of X, and we show that this upper bound is tight for the case of quadratic functions on the hypercube C={0,1}^n, a very well studied case in combinatorial optimization. Using the lower bounds for C we construct a family of globally nonnegative quartic polynomials, which are not sums of squares of rational functions of small degree. To our knowledge this is the first construction for Hilbert's 17th problem of a family of polynomials of bounded degree which need increasing degrees in rational function representations as the number of variables n goes to infinity. We note that representation theory of the symmetric group S_n play a crucial role in our proofs of the lower bounds.

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arXiv2013

Sums of squares and varieties of minimal degree

Let X be a real nondegenerate projective subvariety such that its set of real points is Zariski dense. We prove that every real quadratic form that is nonnegative on X is a sum of squares of linear forms if and only if X is a variety of minimal degree. This substantially extends Hilbert's celebrated characterization of equality between nonnegative forms and sums of squares. We obtain a complete list for the cases of equality and also a classification of the lattice polytopes Q for which every nonnegative Laurent polynomial with support contained in 2Q is a sum of squares.

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arXiv2013

Dimensional Differences Between Faces of the Cones of Nonnegative Polynomials and 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 completely 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 second ordinary power and the second symbolic power of the vanishing ideal of points in projective space.

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arXiv2012

Typical Real Ranks of Binary Forms

We prove a conjecture of Comon and Ottaviani that typical real Waring ranks of bivariate forms of degree $d$ take all integer values between $\lfloor \frac{d+2}{2}\rfloor$ and $d$. That is we show that for all $d$ and all $\lfloor \frac{d+2}{2}\rfloor \leq m \leq d$ there exists a bivariate form $f$ such that $f$ can be written as a linear combination of $m$ $d$-th powers of real linear forms and no fewer, and additionally all forms in an open neighborhood of $f$ also possess this property. Equivalently we show that for all $d$ and any $\lfloor \frac{d+2}{2}\rfloor \leq m \leq d$ there exists a symmetric real bivariate tensor $t$ of order $d$ such that $t$ can be written as a linear combination of $m$ symmetric real tensors of rank 1 and no fewer, and additionally all tensors in an open neighborhood of $t$ also possess this property.

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arXiv2012

Positive Gorenstein Ideals

We introduce positive Gorenstein ideals. These are Gorenstein ideals in the graded ring $\RR[x]$ with socle in degree 2d, which when viewed as a linear functional on $\RR[x]_{2d}$ is nonnegative on squares. Equivalently, positive Gorenstein ideals are apolar ideals of forms whose differential operator is nonnegative on squares. Positive Gorenstein ideals arise naturally in the context of nonnegative polynomials and sums of squares, and they provide a powerful framework for studying concrete aspects of sums of squares representations. We present applications of positive Gorenstein ideals in real algebraic geometry, analysis and optimization. In particular, we present a simple proof of Hilbert's nearly forgotten result on representations of ternary nonnegative forms as sums of squares of rational functions. Drawing on our previous work, our main tools are Cayley-Bacharach duality and elementary convex geometry.

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arXiv2012

Nonnegative Polynomials and Sums of Squares

In the smallest cases where there exist nonnegative polynomials that are not sums of squares we present a complete explanation of this distinction. The fundamental reason that the cone of sums of squares is strictly contained in the cone of nonnegative polynomials is that polynomials of degree $d$ satisfy certain linear relations, known as the Cayley-Bacharach relations, which are not satisfied by polynomials of full degree 2d. For any nonnegative polynomial that is not a sum of squares we can write down a linear inequality coming from a Cayley-Bacharach relation that certifies this fact. We also characterize strictly positive sums of squares that lie on the boundary of the cone of sums of squares and extreme rays of the cone dual to the cone of sums of squares

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arXiv2011

Algebraic boundaries of Hilbert's SOS cones

We study the geometry underlying the difference between non-negative polynomials and sums of squares. The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether-Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized respectively by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

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arXiv2011

ADAM: Analysis of Discrete Models of Biological Systems Using Computer Algebra

Background: Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, Petri nets, and agent-based models, with the goal to gain a better understanding of the system. The computational complexity to analyze the complete dynamics of these models grows exponentially in the number of variables, which impedes working with complex models. Although there exist sophisticated algorithms to determine the dynamics of discrete models, their implementations usually require labor-intensive formatting of the model formulation, and they are oftentimes not accessible to users without programming skills. Efficient analysis methods are needed that are accessible to modelers and easy to use. Method: By converting discrete models into algebraic models, tools from computational algebra can be used to analyze their dynamics. Specifically, we propose a method to identify attractors of a discrete model that is equivalent to solving a system of polynomial equations, a long-studied problem in computer algebra. Results: A method for efficiently identifying attractors, and the web-based tool Analysis of Dynamic Algebraic Models (ADAM), which provides this and other analysis methods for discrete models. ADAM converts several discrete model types automatically into polynomial dynamical systems and analyzes their dynamics using tools from computer algebra. Based on extensive experimentation with both discrete models arising in systems biology and randomly generated networks, we found that the algebraic algorithms presented in this manuscript are fast for systems with the structure maintained by most biological systems, namely sparseness, i.e., while the number of nodes in a biological network may be quite large, each node is affected only by a small number of other nodes, and robustness, i.e., small number of attractors.

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arXiv2009

Convex Forms that are not Sums of Squares

An orbitope is the convex hull of an orbit of a point under the action of a compact group. We derive bounds on volumes of sections of polar bodies of orbitopes, extending our previously developed methods. As an application we realize the cone of convex forms as a section of the cone of nonnegative bi-homogeneous forms and estimate its volume. A convex form has to be nonnegative, but it has not been previously shown that there exist convex forms that are not sums of squares. Combining with the known bounds we show that if the degree is fixed then the cone of convex forms has asymptotically same size as the cone of nonnegative forms and it is significantly larger asymptotically than the cone of sums of squares. This implies existence of convex forms that are not sums of squares, although there are still no known examples.

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arXiv2009

Dimensional Differences between Nonnegative Polynomials and 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. As either the degree or the number of variables grows the gaps become very large, asymptotically the gaps approach the full dimension of the vector space of polynomials in $n$ variables of degree 2d. The gaps occur generically, they are not a product of selecting special faces of the cones. Using these dimensional differences we show how to derive inequalities that separate nonnegative polynomials from sums of squares; the inequalities will hold for all sums of squares, but will fail for some nonnegative polynomials.

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arXiv2008

Experimental study of energy-minimizing point configurations on spheres

In this paper we report on massive computer experiments aimed at finding spherical point configurations that minimize potential energy. We present experimental evidence for two new universal optima (consisting of 40 points in 10 dimensions and 64 points in 14 dimensions), as well as evidence that there are no others with at most 64 points. We also describe several other new polytopes, and we present new geometrical descriptions of some of the known universal optima.

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arXiv2004

Volumes of Nonnegative Polynomials, Sums of Squares and Powers of Linear Forms

We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient dimension) of compact sections of the three cones. We show that the bounds are asymptotically exact if the degree is fixed and number of variables tends to infinity. When the degree is larger than two it follows that there are significantly more non-negative polynomials than sums of squares and there are significantly more sums of squares than sums of powers of linear forms. Moreover, we quantify the exact discrepancy between the cones; from our bounds it follows that the discrepancy grows as the number of variables increases.

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arXiv2003

Convex Geometry of Orbits

We study metric properties of convex bodies B and their polars B^o, where B is the convex hull of an orbit under the action of a compact group G. Examples include the Traveling Salesman Polytope in polyhedral combinatorics (G=S_n, the symmetric group), the set of non-negative polynomials in real algebraic geometry (G=SO(n), the special orthogonal group), and the convex hull of the Grassmannian and the unit comass ball in the theory of calibrated geometries (G=SO(n), but with a different action). We compute the radius of the largest ball contained in the symmetric Traveling Salesman Polytope, give a reasonably tight estimate for the radius of the Euclidean ball containing the unit comass ball and review (sometimes with simpler and unified proofs) recent results on the structure of the set of non-negative polynomials (the radius of the inscribed ball, volume estimates, and relations to the sums of squares). Our main tool is a new simple description of the ellipsoid of the largest volume contained in B^o.

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arXiv2003

There are Significantly More Nonnegative Polynomials than Sums of Squares

We investigate the quantitative relationship between nonnegative polynomials and sums of squares of polynomials. We show that if the degree is fixed and the number of variables grows then there are significantly more nonnegative polynomials than sums of squares. More specifically, we take compact bases of the cone of nonnegative polynomials and the cone of sums of squares and derive bounds for the volumes of the bases. If the degree is greater than 2 then we show that the ratio of the volumes of the bases, raised to the power reciprocal to the ambient dimension, tends to 0 as the number of variables tends to infinity.

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arXiv2002

Convexity properties of the cone of nonnegative polynomials

We study metric properties of the cone of homogeneous non-negative multivariate polynomials and the cone of sums of powers of linear forms, and the relationship between the two cones. We compute the maximum volume ellipsoid of the natural base of the cone of non-negative polynomials and the minimum volume ellipsoid of the natural base of the cone of powers of linear forms and compute the coefficients of symmetry of the bases. The multiplication by (x_1^2 + ... + x_n^2)^m induces an isometric embedding of the space of polynomials of degree $2k$ into the space of polynomials of degree 2(k+m), which allows us to compare the cone of non-negative polynomials of degree $2k$ and the cone of sums of 2(k+m)-powers of linear forms. We estimate the volume ratio of the bases of the two cones and the rate at which it approaches 1 as m grows.

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