Real Algebraic and Analytic Geometry |

Closures of quadratic modules.

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Submission: 2009, April 9.

*Abstract:
We consider the problem of determining the closure $\overline{M}$
of a quadratic module $M$ in a commutative $\Bbb{R}$-algebra with
respect to the finest locally convex topology. This is of interest
in deciding when the moment problem is solvable \cite{S1}
\cite{S2} and in analyzing algorithms for polynomial optimization
involving semidefinite programming \cite{L}. The closure of a
semiordering is also considered, and it is shown that the space
$\mathcal{Y}_M$ consisting of all semiorderings lying over $M$
plays an important role in understanding the closure of $M$. The
result of Schm\"udgen for preorderings in \cite{S2} 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 $\overline{M}$ which is valid in many
cases.*

Mathematics Subject Classification (2000): 12D15, 14P99, 44A60.

Keywords and Phrases: moment problem, positive polynomials, sums of squares.

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