Closures of quadratic modules

TL;DR

This paper investigates the closure of quadratic modules in commutative algebras, extending existing theorems to better understand the moment problem and polynomial optimization, with implications for semidefinite programming.

Contribution

It extends Schmuedgen's fiber theorem to quadratic modules and provides a recursive method to describe their closure in various cases.

Findings

Extended the fiber theorem for quadratic modules.

Constructed a non-archimedean quadratic module with the strong moment property.

Provided a recursive description of the closure of quadratic modules.

Abstract

We consider the problem of determining the closure of a quadratic module M in a commutative R-algebra with respect to the finest locally convex topology. This is of interest in deciding when the moment problem is solvable and in analyzing algorithms for polynomial optimization involving semidefinite programming. The closure of a semiordering is also considered, and it is shown that the space of all semiorderings lying over M plays an important role in understanding the closure of M. The fibre theorem of Schmuedgen for preorderings is strengthened and extended to quadratic modules. The extended result is used to construct an example of a non-archimedean quadratic module describing a compact semialgebraic set that has the strong moment property. The same result is used to obtain a recursive description of the closure of M which is valid in many cases.