Solving moment problems by dimensional extension

In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of [17] are efficient and numerically stable for computing complex roots, algorithms based on moment matrices [12] allow the incorporation of...

The fibre theorem \cite{schm2003} for the moment problem on closed semi-algebraic subsets of $\R^d$ is generalized to finitely generated real unital algebras. As an application two new theorems on the rational multidimensional moment problem are proved. Another application is a characterization of moment functionals on the polynomial algebra $\R[x_1,\dots,x_d]$ in terms of extensions. Finally, the fibre theorem and the extension theorem are used to reprove basic results on th...

We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence $y$, we show that $y$ lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set $K \subseteq \re^n$ if and only if the associated Riesz functional $L_y$ is $K$-positive. For a determining set $K$, we prove that if $L_y$ is str...

In this paper we improve the bounds for the Carath\'eodory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower and upper bounds on algebraic varieties, $\mathbb{R}^n$, and $[0,1]^n$. We also treat moment problems with small gaps. We find that for every $\varepsilon>0$ and $d\in\mathbb{N}$ there is a $n\in\mathbb{N}$ such that we can construct a moment functional $L:\mathbb{R}[x_1,\dots,x_n]_{\leq d}\rightarr...

We present a new algorithm for computing the real radical of an ideal and, more generally, the-radical of, which is based on convex moment optimization. A truncated positive generic linear functional vanishing on the generators of is computed solving a Moment Optimization Problem (MOP). We show that, for a large enough degree of truncation, the annihilator of generates the real radical of. We give an effective, general stopping criterion on the degree to detect when the prime...

The fibre theorem \cite{schm2003} for the moment problem on closed semi-algebraic subsets of $\R^d$ is generalized to finitely generated real unital algebras. A charaterization of moment functionals on the polynomial algebra $\R[x_1,\dots,x_d]$ in terms of extensions is given. These results are applied to reprove basic results on the complex moment problem due to Stochel and Szafraniec \cite{stochelsz} and Bisgaard \cite{bisgaard}.

This paper studies Positivstellens\"atze and moment problems for sets that are given by universal quantifiers. Let $Q$ be a closed set and let $g = (g_1,...,g_s)$ be a tuple of polynomials in two vector variables $x$ and $y$. Then $K$ is described as the set of all points $x$ such that each $g_j(x, y) \ge 0$ for all $y \in Q$. Fix a measure $\nu$ with $supp(\nu) = Q$, and assume it satisfies the Carleman condition. The first main result of the paper is a Positivstellensatz ...

For a degree 2n finite sequence of real numbers $\beta \equiv \beta^{(2n)}= \{ \beta_{00},\beta_{10}, \beta_{01},\cdots, \beta_{2n,0}, \beta_{2n-1,1},\cdots, \beta_{1,2n-1},\beta_{0,2n} \}$ to have a representing measure $\mu $, it is necessary for the associated moment matrix $\mathcal{M}(n)$ to be positive semidefinite, and for the algebraic variety associated to $\beta $, $\mathcal{V}_{\beta} \equiv \mathcal{V}(\mathcal{M}(n))$, to satisfy $\operatorname{rank} \mathcal{M}(...

We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a basic semi-algebraic set K is convex but its defining polynomials are not, we provide...

We show that the closed convex hull of any one-dimensional semi-algebraic subset of R^n has a semidefinite representation, meaning that it can be written as a linear projection of the solution set of some linear matrix inequality. This is proved by an application of the moment relaxation method. Given a nonsingular affine real algebraic curve C and a compact semialgebraic subset K of its R-points, the preordering P(K) of all regular functions on C that are nonnegative on K is...