Representation of nonnegative convex polynomials

Nonnegative Morse polynomial functions and polynomial optimization

In this paper we study the representation of Morse polynomial functions which are nonnegative on a compact basic closed semi-algebraic set in $\mathbb R^n$, and having only finitely many zeros in this set. Following C. Bivià-Ausina, we introduce two classes of non-degenerate polynomials for which the algebraic sets defined by them are compact. As a consequence, we study the representation of nonnegative Morse polynomials on these kinds of non-degenerate algebraic sets. Moreover, we apply these results to study the polynomial optimization problem for Morse polynomial functions.

充分正则化的非负四次多项式是平方和

Sufficiently Regularized Nonnegative Quartic Polynomials are Sum-of-Squares

研究揭示了正则化下非负四次多项式成为平方和的条件，指出四次正则化在多项式优化中的关键作用，区分了可计算与不可计算的优化区域。

Approximating Nonnegative Polynomials via Spectral Sparsification

We study polyhedral approximations to the cone of nonnegative polynomials. We show that any constant ratio polyhedral approximation to the cone of nonnegative degree $2d$ forms in $n$ variables has to have exponentially many facets in terms of $n$. We also showthat for fixed $m \geq 3$, all linear $m$ dimensional sections of the nonnegative cone that include $(x_1^2+x_2^2+\ldots + x_n^2)^d$ has a costant ratio polyhedral approximation with $O(n^{m-2})$ many facets. Our approach is convex geometric, and parts of the argument rely on the recent solution of Kadison-Singer problem. We also discuss…

A sum of squares approximation of nonnegative polynomials

We show that every real nonnegative polynomial $f$ can be approximated as closely as desired by a sequence of polynomials $\{f_ε\}$ that are sums of squares. Each $f_ε$ has a simple et explicit form in terms of $f$ and $ε$. A special representation is also obtained for convex polynomials, nonnegative on a convex semi-algebraic set.

Nonnegative polynomials and their Carathéodory number

In 1888 Hilbert showed that every nonnegative homogeneous polynomial with real coefficients of degree $2d$ in $n$ variables is a sum of squares if and only if $d=1$ (quadratic forms), $n=2$ (binary forms) or $(n,d)=(3,2)$ (ternary quartics). In these cases, it is interesting to compute canonical expressions for these decompositions. Starting from Carathéodory's Theorem, we compute the Carathéodory number of Hilbert cones of nonnegative quadratic and binary forms.

Separating inequalities for nonnegative polynomials that are not sums of squares

Ternary sextics and quaternary quartics are the smallest cases where there exist nonnegative polynomials that are not sums of squares (SOS). A complete classification of the difference between these cones was given by G. Blekherman via analyzing the extreme rays of the corresponding dual cones. However, an exact computational approach in order to build separating extreme rays for nonnegative polynomials that are not sums of squares is a widely open problem. We provide a method substantially simplifying this computation for certain classes of polynomials on the boundary of the PSD cones. In par…

Optimization over the Boolean Hypercube via Sums of Nonnegative Circuit Polynomials

Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems are based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate the analysis of optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS based certificates remain valid: First, for polynomials, which are nonnegative over the $n$…

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 a…

Lectures on Nonnegative Polynomials and Sums of Squares

These lecture notes provide an informal introduction to the theory of nonnegative polynomials and sums of squares. We highlight the history and some recent developments, especially the new connections with classical (complex) algebraic geometry and commutative algebra. We briefly discuss the connection to semidefinite programming and applications. We expect the reader to be familiar with algebra and fundamentals of algebraic geometry, on the level of a graduate course, and know basic facts about positive semidefinite matrices.

SOS approximations of nonnegative polynomials via simple high degree perturbation

We show that every real polynomial $f$ nonnegative on $[-1,1]^{n}$ can be approximated in the $l_{1}$-norm of coefficients, by a sequence of polynomials $\{f_{\ep r}\}$ that are sums of squares. This complements the existence of s.o.s. approximations in the denseness result of Berg, Christensen and Ressel, as we provide a very simple and \textit{explicit} approximation sequence. Then we show that if the Moment Problem holds for a basic closed semi-algebraic set $K_S\subset\R^n$ with nonempty interior, then every polynomial nonnegative on $K_S$ can be approximated in a similar fashion by elemen…

Sums of Separable and Quadratic Polynomials

We study separable plus quadratic (SPQ) polynomials, i.e., polynomials that are the sum of univariate polynomials in different variables and a quadratic polynomial. Motivated by the fact that nonnegative separable and nonnegative quadratic polynomials are sums of squares, we study whether nonnegative SPQ polynomials are (i) the sum of a nonnegative separable and a nonnegative quadratic polynomial, and (ii) a sum of squares. We establish that the answer to question (i) is positive for univariate plus quadratic polynomials and for convex SPQ polynomials, but negative already for bivariate quarti…

Nonnegative polynomials from vector bundles on real curves

We observe that the E-resultant of a very ample rank 2 vector bundle E on a real projective curve (with no real points) is nonnegative when restricted to the space of real sections. Moreover, we show that if E has a section vanishing at exactly two points and the degree d of E satisfies d(d-6)> 4g-5, then this polynomial cannot be written as a sum of squares.

Duality of sum of nonnegative circuit polynomials and optimal SONC bounds

Circuit polynomials are polynomials satisfying a number of conditions that make it easy to compute sharp and certifiable global lower bounds for them. Consequently, one may use them to find certifiable lower bounds for any polynomial by writing it as a sum of circuit polynomials with known lower bounds (if possible), in a fashion similar to the better-known sum-of-squares polynomials. Seidler and de Wolff recently showed that sums of nonnegative circuit (SONC) polynomials can be used to compute global lower bounds (called SONC bounds) for polynomials in this manner in polynomial time, as long …

A Generalized Muirhead Inequality and Symmetric Sums of Nonnegative Circuits

Circuit polynomials are a certificate of nonnegativity for real polynomials, which can be derived via a generalization of the classical inequality of arithmetic and geometric means. In this article, we show that similarly nonnegativity of symmetric real polynomials can be certified via a generalization of the classical Muirhead inequality. Moreover, we show that a nonnegative symmetric polynomial admits a decomposition into sums of nonnegative circuit polynomials if and only if it satisfies said generalized Muirhead condition. The latter re-proves a result by Moustrou, Naumann, Riener, Theobal…

多项式在某些仿射超曲面非负部分上的非负性

Positivity of polynomials on the nonnegative part of certain affine hypersurfaces

研究多项式在特定仿射超曲面非负部分上的非负性，证明严格正多项式可由正系数多项式表示，推广Pólya定理。

Polynomials nonnegative on the cylinder

In 2010, Marshall settled the strip conjecture, according to which every polynomial in $\mathbb{R}[x,y]$, nonnegative on the strip $[-1,1]\times\mathbb{R}$, is a sum of squares and of squares times $1-x^2$. We consider affine nonsingular curves $C$ over $\mathbb{R}$ with $C(\mathbb{R})$ compact, and study the question whether every $f$ in $\mathbb{R}[C][y]$, nonnegative on $C(\mathbb{R})\times\mathbb{R}$, is a sum of squares in $\mathbb{R}[C][y]$. We give an affirmative answer under the condition that $f$ has only finitely many zeros in $C(\mathbb{R})\times\mathbb{R}$. For $C$ the circle $x_1^…

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 var…

非负多项式与代数曲线上的矩问题

Nonnegative Polynomials and Moment Problems on Algebraic Curves

本文研究了代数曲线上的非负多项式锥的面结构及其对偶矩锥的Carathéodory数，揭示了拓扑对四次规则复杂性的影响，并应用于截断矩问题。

Multihomogenous Nonnegative Polynomials and Sums of Squares

We refine and extend quantitative bounds, on the fraction of nonnegative polynomials that are sums of squares, to the multihomogenous case.

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 ca…

Amoebas, Nonnegative Polynomials and Sums of Squares Supported on Circuits

We completely characterize sections of the cones of nonnegative polynomials, convex polynomials and sums of squares with polynomials supported on circuits, a genuine class of sparse polynomials. In particular, nonnegativity is characterized by an invariant, which can be immediately derived from the initial polynomial. Furthermore, nonnegativity of such polynomials $f$ coincides with solidness of the amoeba of $f$, i.e., the Log-absolute-value image of the algebraic variety $\mathcal{V}(f) \subset (\mathbb{C}^*)^n$ of $f$. These results generalize earlier works both in amoeba theory and real al…

S.o.s. approximation of polynomials nonnegative on a real algebraic set

With every real polynomial $f$, we associate a family $\{f_{εr}\}_{ε, r}$ of real polynomials, in explicit form in terms of $f$ and the parameters $ε>0,r\in N$, and such that $\Vert f-f_{εr}\Vert_1\to 0$ as $ε\to 0$. Let $V\subset R^n$ be a real algebraic set described by finitely many polynomials equations $g_j(x)=0,j\in J$, and let $f$ be a real polynomial, nonnegative on $V$. We show that for every $ε>0$, there exist nonnegative scalars $\{λ_j(ε)\}_{j\in J}$ such that, for all $r$ sufficiently large, $$f_{εr}+\sum_{j\in J} λ_j(ε) g_j^2,\quad is a sum of squares.$$ This representation is an …

Lower Bounds for Polynomials with Simplex Newton Polytopes Based on Geometric Programming

In this article, we propose a geometric programming method in order to compute lower bounds for real polynomials. We provide new sufficient conditions for polynomials to be nonnegative as well as to have a sum of binomial squares representation. These criteria rely on the coefficients and the support of a polynomial and generalize all previous ones by Lasserre, Ghasemi, Marshall, Fidalgo and Kovacec to polynomials with arbitrary simplex Newton polytopes. This generalization yields a geometric programming approach for computing lower bounds for polynomials that significantly extends the geometr…

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…

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 li…

Isoparametric polynomials and sums of squares

Hilbert's 17th problem asks that whether every nonnegative polynomial can be a sum of squares of rational functions. It has been answered affirmatively by Artin. However, the question as to whether a given nonnegative polynomial is a sum of squares of polynomials is still a central question in real algebraic geometry. In this paper, we solve this question completely for the nonnegative polynomials associated with isoparametric polynomials, initiated by E. Cartan, which define the focal submanifolds of the corresponding isoparametric hypersurfaces.

Refined Asymptotics for Multigraded Sums of Squares

To prove that a polynomial is nonnegative on R^n one can try to show that it is a sum of squares of polynomials (SOS). The latter problem is now known to be reducible to a semidefinite programming (SDP) computation much faster than classical algebraic methods, thus enabling new speed-ups in algebraic optimization. However, exactly how often nonnegative polynomials are in fact sums of squares of polynomials remains an open problem. Blekherman was recently able to show that for degree k polynomials in n variables -- with k>=4 fixed -- those that are SOS occupy a vanishingly small fraction of tho…

Power mean inequalities and sums of squares

For fixed degree and increasing number of variables the dimension of the vector space of $n$-variate real symmetric homogeneous polynomials (forms) of degree $d$ stabilizes. We study the limits of the cones of symmetric nonnegative polynomials and symmetric sums of squares, when expressed in power-mean or monomial-mean basis. These limits correspond to forms with stable expression in power-mean (or monomial-mean) polynomials that are globally nonnegative (resp. sums of squares) regardless of the number of variables. We introduce partial symmetry reduction to describe the limit cone of symmetri…

矩、平方和与热带化

Moments, Sums of Squares, and Tropicalization

本文用热带化刻画半代数集上矩锥与伪矩锥的组合结构，借此区分非负多项式与平方和，并揭示平方和逼近的局限。

Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets

A set $S\subseteq \re^n$ is called to be {\it Semidefinite (SDP)} representable if $S$ equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). The contributions of this paper are: (i) For bounded SDP representable sets $W_1,...,W_m$, we give an explicit construction of an SDP representation for $\cv{\cup_{k=1}^mW_k}$. This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set $S$, we prove sufficient condition: the boundary $\bdS$ i…

顽固多项式

Stubborn Polynomials

研究顽固多项式在实代数几何中的性质，证明三元六次多项式非负当且仅当其实delta不变量至少为9。

非负卦限上多项式优化的可处理凸松弛层级

Tractable hierarchies of convex relaxations for polynomial optimization on the nonnegative orthant

针对非负卦限半代数集上的多项式优化问题，提出基于Pólya正性定理扩展的可调节矩阵尺寸的半定松弛层级，证明其收敛速率，且在两类应用中效果与速度均优于标准Moment-SOS层级。

Total nonnegativity and stable polynomials

We consider homogeneous multiaffine polynomials whose coefficients are the Plücker coordinates of a point $V$ of the Grassmannian. We show that such a polynomial is stable (with respect to the upper half plane) if and only if $V$ is in the totally nonnegative part of the Grassmannian. To prove this, we consider an action of matrices on multiaffine polynomials. We show that a matrix $A$ preserves stability of polynomials if and only if $A$ is totally nonnegative. The proofs are applications of classical theory of totally nonnegative matrices, and the generalized Pólya-Schur theory of Borcea and…

Multiplication of a Schubert polynomial by a Stanley symmetric polynomial

We prove, combinatorially, that the product of a Schubert polynomial by a Stanley symmetric polynomial is a truncated Schubert polynomial. Using Monk's rule, we derive a nonnegative combinatorial formula for the Schubert polynomial expansion of a truncated Schubert polynomial. Combining these results, we give a nonnegative combinatorial rule for the product of a Schubert and a Schur polynomial in the Schubert basis.

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), wh…

Sums of Squares and Sparse Semidefinite Programming

We consider two seemingly unrelated questions: the relationship between nonnegative polynomials and sums of squares on real varieties, and sparse semidefinite programming. This connection is natural when a real variety $X$ is defined by a quadratic square-free monomial ideal. In this case nonnegative polynomials and sums of squares on $X$ are also natural objects in positive semidefinite matrix completion. Nonnegative quadratic forms over $X$ naturally correspond to partially specified matrices where all of the fully specified square blocks are PSD, and sums of squares quadratic forms naturall…

Extreme Nonnegative Quadratics over Stanley Reisner Varieties

We consider the convex geometry of the cone of nonnegative quadratics over Stanley-Reisner varieties. Stanley-Reisner varieties (which are unions of coordinate planes) are amongst the simplest real projective varieties, so this is potentially a starting point that can generalize to more complicated real projective varieties. This subject has some suprising connections to algebraic topology and category theory, which we exploit heavily in our work. These questions are also valuable in applied math, because they directly translate to questions about positive semidefinite (PSD) matrices. In parti…

Hyperbolic polynomials, interlacers, and sums of squares

Hyperbolic polynomials are real polynomials whose real hypersurfaces are nested ovaloids, the inner most of which is convex. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. Here we investigate the special connection between a hyperbolic polynomial and the set of polynomials that interlace it. This set of interlacers is a convex cone, which we write as a linear slice of the cone of nonnegative polynomials. In particular, this allows us to realize any hyperbolicity cone as a slice of the cone of nonnegative polynomials. Usi…

$\u0001d4a2$-洛伦兹多项式

$\mathcal{K}$-Lorentzian Polynomials

本文研究正常凸锥上的𝓚-洛伦兹多项式，揭示其与自对偶锥𝓚正线性映射的关联，发现二次𝓚-洛伦兹多项式集随锥变化难以算法刻画，并证明其与𝓚-完全对数凹多项式等价。

Positive trigonometric polynomials

We study the boundary of the nonnegative trigonometric polynomials from the algebraic point of view. In particularly, we show that it is a subset of an irreducible algebraic hypersurface and established its explicit form in terms of resultants.