Universität Leipzig（莱比锡大学）；Georgia Institute of Technology（佐治亚理工学院）；Université Côte d’Azur（蔚蓝海岸大学）；Technische Universität Dortmund（多特蒙德工业大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Lorenzo Baldi, Grigoriy Blekherman, Khazhgali Kozhasov, Daniel Plaumann, Bruce Reznick, Rainer Sinn
The relationship between nonnegative polynomials and sums of squares is a classical topic in real algebraic geometry. We study \emph{stubborn polynomials} $f$ on a real variety $X$, which are polynomials nonnegative on $X$, such that no odd power of $f$ is a sum of squares. Previously, stubborn polynomials were studied only in the globally nonnegative case, with results restricted to polynomials nonnegative on $\mathbb{P}^2$. We fully characterize stubborn polynomials on smooth curves, showing that a polynomial on a smooth totally real curve is stubborn if and only if all of its zeros are real. This implies that there exist smooth curves with no stubborn polynomials in low degree, while stubborn polynomials must exist in sufficiently high degrees on curves with positive genus. We explore the much more delicate situation with singular and reducible curves. While being real-rooted always implies being stubborn, there also exist singular curves with no stubborn polynomials at all.
We then analyze the case of ternary sextics, i.e.,~polynomials of degree $6$ on $\mathbb{P}^2$. We prove the Conjecture of Blekherman, Kozhasov, and Reznick that a nonnegative ternary sextic is stubborn if and only if its real delta-invariant is at least 9. To analyze this case, we develop results for lifting stubborn polynomials from curves to higher dimensional varieties and use the theory of weak Del Pezzo surfaces. We complement these results with structural properties of stubborn polynomials and present many explicit examples.
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.
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.
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…
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 $x_1^…
S.o.s. approximation of polynomials nonnegative on a real algebraic set
对于每一个实多项式$f$，我们关联一个实多项式族$\{f_{εr}\}_{ε, r}$，以$f$和参数$ε>0,r\in N$的显式形式，并且使得$\Vert f-f_{εr}\Vert_1\to 0$为$ε\to 0$。设$V\subset R^n$是由多个多项式方程$g_j(x)=0,j\in J$描述的实代数集，$f$是$V$上的非负实多项式.我们发现，对于每一个$ε>0$，存在非负标量$\{λ_j(ε)\}_{j\in J}$，这样，对于所有的$r$足够大，$$f_{εr}+\sum_{j\in J} λ_j(ε) g_j^2,\quad is a sum of squares.$$这种表示是一个明显的证书的非负性的$f_{εr}$的$V$，非常具体的$g_j$定义的$V$集。特别是，它是有效的，与{ \it否}的$V$假设。此外，从计算的角度来看，这种表示也是有用的，因为我们可以定义半定编程松弛，以近似在一个真正的代数集$V$或半代数集$K$上的$f$的全局最小值，并再次在$V$或$K$上具有{ \it no}假设。
The concept of sums of nonnegative circuit polynomials (SONC) was recently introduced as a new certificate of nonnegativity especially for sparse polynomials. In this paper, we explore the relationship between nonnegative polynomials and SONC polynomials. As a first result, we provide sufficient conditions for nonnegative polynomials with general Newton polytopes to be SONC polynomials, which generalizes the previous result on nonnegative polynomials with simplex Newton polytopes. Secondly, we prove that every SONC polynomial admits a SONC decomposition without cancellation. In other words, SO…
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…
In this article we combine two developments in polynomial optimization. On the one hand, we consider nonnegativity certificates based on sums of nonnegative circuit polynomials, which were recently introduced by the second and the third author. On the other hand, we investigate geometric programming methods for constrained polynomial optimization problems, which were recently developed by Ghasemi and Marshall. We show that the combination of both results yields a new method to solve certain classes of constrained polynomial optimization problems. We test the new method experimentally and compa…
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…
Recently, the second and the third author developed sums of nonnegative circuit polynomials (SONC) as a new certificate of nonnegativity for real polynomials, which is independent of sums of squares. In this article we show that the SONC cone is full-dimensional in the cone of nonnegative polynomials. We establish a Positivstellensatz which guarantees that every polynomial which is positive on a given compact, semi-algebraic set can be represented by the constraints of the set and SONC polynomials. Based on this Positivstellensatz we provide a hierarchy of lower bounds converging against the m…
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…
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.
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…
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 $n$…
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…
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.
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…
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 …
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…
Tractable hierarchies of convex relaxations for polynomial optimization on the nonnegative orthant
Institute of Mathematics, Vietnam Academy of Science and Technology（越南科技科学院数学研究所）；CNRS（法国国家科学研究中心（CNRS）、劳伦斯·亚当斯研究所（LAAS））；LAAS（图卢兹大学；劳伦斯·亚当斯研究所）；Université de Toulouse（新加坡国立大学数学系）；LAAS；Department of Mathematics, National University of Singapore
The relationship between nonnegative polynomials and sums of squares is one of the central questions in real algebraic geometry. A modern approach is to look at nonnegative polynomials and sums of squares on a real variety. We survey the rich history of this area and focus on recent developments highlighting unexpected links to complex algebraic geometry and commutative algebra.
A real polynomial $f$ is called local nonnegative at a point $p$, if it is nonnegative in a neighbourhood of $p$. In this paper, a sufficient condition for determining this property is constructed. Newton's principal part of $f$ (denoted as $f_N$) plays a key role in this process. We proved that if every $F$-face, $(f_N)_F$, of $f_N$ is strictly positive over $(\mathbb{R}\setminus 0)^n$, then $f$ is local nonnegative at the origin $O$.
A set $S\subset \mathbb{R}^n$ is a nonnegativity witness for a set $U$ of real homogeneous polynomials if $F$ in $U$ is nonnegative on $\mathbb{R}^n$ if and only if it is nonnegative at all points of $S$. We prove that the union of the hyperplanes perpendicular to the elements of a root system $Φ\subseteq \mathbb{R}^n$ is a witness set for nonnegativity of forms of low degree which are invariant under the reflection group defined by $Φ$. We prove that our bound for the degree is sharp for all reflection groups which contain multiplication by $-1$. We then characterize subspaces of forms of arb…
We provide a complete and explicit characterization of the real zeros of sums of nonnegative circuit (SONC) polynomials, a recent certificate for nonnegative polynomials independent of sums of squares. As a consequence, we derive an exact determination of the number $B''_{n+1,2d}$ for all $n$ and $d$. $B''_{n+1,2d}$ is defined to be the supremum of the number of zeros of all homogeneous $n+1$-variate polynomials of degree $2d$ in the SONC cone. The analogously defined numbers $B_{n+1,2d}$ and $B'_{n+1,2d}$ for the nonnegativity cone and the cone of sums of squares were first introduced and stu…
