Related papers: Sums of Squares on the Hypercube

The problem of characterizing a real polynomial $f$ as a sum of squares of polynomials on a real algebraic variety $V$ dates back to the pioneering work of Hilbert in [Mathematische Annalen 32.3 (1888): 342-350]. In this paper, we…

We study the problem of representing multivariate polynomials with rational coefficients, which are nonnegative and strictly positive on finite semialgebraic sets, using rational sums of squares. We focus on the case of finite semialgebraic…

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…

In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex)…

We present an example of a strictly positive polynomial with rational coefficients that can be decomposed as a sum of squares of polynomials over $\R$ but not over $\Q$. This answers an open question by C. Scheiderer posed as the second…

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…

A polynomial that is nonnegative need not be a sum of squares of polynomials. This classical gap, identified by Hilbert in 1888, lies at the heart of why the global optimization of multivariate quartic polynomials is NP-hard. Yet we show…

We ask whether every polynomial function that is non-negative on a real algebraic curve can be expressed as a sum of squares in the coordinate ring. Scheiderer has classified all irreducible curves for which this is the case. For reducible…

A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares. We show more generally that every nonnegative quadratic form on a real projective variety $X$ of minimal degree is a sum of…

This article deals with a quantitative aspect of Hilbert's seventeenth problem: producing a collection of real polynomials in two variables of degree 8 in one variable which are positive but are not a sum of three squares of rational…

We consider min-max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables.…

The paper proves sum-of-square-of-rational-function based representations (shortly, sosrf-based representations) of polynomial matrices that are positive semidefinite on some special sets: $\mathbb{R}^n;$ $\mathbb{R}$ and its intervals…

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…

Let X be a projective cubic hypersurface of dimension 11 or more, which is defined over the rationals. In this paper it is shown that X contains rational points provided that the cubic form defining X can be written as the sum of two forms…

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…

We study the sums of squares on cylinders of the form $X \times \mathbb{A}_K$ for a (weakly) factorial curve $C$. We prove the equality of the Pythagoras numbers of the ring of regular functions on the cylinder with that of the field of…

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…

The Positivstellens\"atze of Putinar and Schm\"udgen show that any polynomial $f$ positive on a compact semialgebraic set can be represented using sums of squares. Recently, there has been large interest in proving effective versions of…

We study the problem of decomposing a polynomial $p$ into a sum of $r$ squares by minimizing a quadratically penalized objective $f_p(\mathbf{u}) = \left\lVert \sum_{i=1}^r u_i^2 - p\right\lVert^2$. This objective is nonconvex and is…

Consider the optimization problem $p_{\min, Q} := \min_{\mathbf{x} \in Q} p(\mathbf{x})$, where $p$ is a degree $m$ multivariate polynomial and $Q := [0, 1]^n$ is the hypercube. We provide explicit degree and error bounds for the sums of…