## D. Henrion - S. Naldi - M. Safey El Din

# Algebraic certificates for the truncated moment problem

created by naldi on 20 Jul 2023

[

BibTeX]

*preprint*

**Inserted:** 20 jul 2023

**Last Updated:** 20 jul 2023

**Year:** 2023

**Abstract:**

The truncated moment problem consists of determining whether a given
finitedimensional vector of real numbers y is obtained by integrating a basis
of the vector space of polynomials of bounded degree with respect to a
non-negative measure on a given set K of a finite-dimensional Euclidean space.
This problem has plenty of applications e.g. in optimization, control theory
and statistics. When K is a compact semialgebraic set, the duality between the
cone of moments of non-negative measures on K and the cone of non-negative
polynomials on K yields an alternative: either y is a moment vector, or y is
not a moment vector, in which case there exists a polynomial strictly positive
on K making a linear functional depending on y vanish. Such a polynomial is an
algebraic certificate of moment unrepresentability. We study the complexity of
computing such a certificate using computer algebra algorithms.