Bounded Polynomials, Sums of Squares, and the Moment Problem

Plaumann, Daniel

Bounded Polynomials, Sums of Squares, and the Moment Problem

Files

Diss_Plaumann.pdf(2.89 MB)

Date

2008

Title in another language

Beschränkte Polynome, Quadratsummen und das Momentenproblem

Publication type

Dissertation

Abstract

We study representations of positive polynomial functions as sums of squares or weighted sums of squares (in so-called preorderings). Approximating positive polynomials by such representations is related to the moment problem from functional analysis. In particular, we study the geometry of bounded polynomial functions in connection with a recent theorem of Schmüdgen that gives a necessary and sufficient condition for the moment problem on a semialgebraic set in terms of the moment problem on the fibres of a bounded polynomial map.

In Chapter 1, we show how the ring of bounded polynomial functions on a semialgebraic set can (under suitable conditions) be represented as the ring of regular functions on a quasiprojective variety. As an application, we use a theorem of Zariski to show that the ring of bounded functions on a sufficiently regular two-dimesional set is a finitely generated R-algebra.

Chapter 2 contains references to known results about positivity and sums of squares. In particular, we prove a stronger version of the Powers-Scheiderer theorem concerning the existence of degree bounds for preorderings.

Chapter 3 is devoted to the case of curves. Here, we generalise results of Scheiderer from the irreducible to the reducible case. We classify all real curves having the moment property. We also prove a result concerning the extension of psd functions from curves to ambient space.

In Chapter 4, we apply the techniques from the previous chapters to the moment problem and sums of squares in the two-dimensional case. We also reprove a theorem of Roggero concerning the divisor class group of a real variety.

Summary in another language

Wir studieren Darstellungen positiver Polynomfunktionen als Summen von Quadraten oder gewichtete Summen von Quadraten (in sog. Präordnungen). Die Approximation positiver Polynome durch solche Darstellungen steht im Zusammenhang mit dem Momentenproblem der Funktionalanalysis. Insbesondere studieren wir die Geometrie beschränkter Polynomfunktionen im Kontext eines Satzes von Schmüdgen, der eine notwendige und hinreichende Bedingung für die Lösbarkeit des Momentenproblems für semialgebraische Mengen durch das Momentenproblem auf den Fasern einer beschränkten Polynomabbildung gibt.

In Kapitel 1 zeigen wir, wie der Ring der beschränkten Polynomfunktionen auf einer semialgebraischen Menge (unter geeigneten Voraussetzungen) als Ring der regulären Funktionen einer quasi-projektiven Varietät aufgefasst werden kann. Dies wenden wir zusammen mit einem Resultat von Zariski an, um zu beweisen, dass der Ring der beschränkten Funktionen einer hinreichend regulären zwei-dimensionalen Menge stets eine endlich-erzeugte R-Algebra ist.

Kapitel 2 gibt Referenzen zu bekannten Resultaten über Positivität und Quadratsummen. Insbesondere beweisen wir eine stärkere Fassung eines Satzes von Powers und Scheiderer über die Existenz von Gradschranken für Präordnungen.

Kapitel 3 ist dem Fall von Kurven gewidmet. Wir verallgemeinern Scheiderers Resultate über irreduzible Kurven auf den reduziblen Fall. Wir klassifizieren alle reelle Kurven, welche die Momenteneigenschaft besitzten. Weiter beweisen wir einen Satz über die Fortsetzbarkeit positiver Funktionen von Kurven auf den umgebenden Raum.

In Kapitel 4 wenden wir die zuvor gewonnenen Resultate im Fall von Flächen an. Wir geben auch einen neuen Beweis eines Satzes von Roggero über die Divisorklassengruppe einer reellen Varietät.

Subject (DDC)

510 Mathematics

Keywords

Quadratsummen,Summen von Quadraten,Holomorphiering,Beschränkte Polynome,Sums of Squares,holomorphy ring,bounded polynomials

Cite This

ISO 690

PLAUMANN, Daniel, 2008. Bounded Polynomials, Sums of Squares, and the Moment Problem [Dissertation]. Konstanz: University of Konstanz

BibTex

@phdthesis{Plaumann2008Bound-740,
  year={2008},
  title={Bounded Polynomials, Sums of Squares, and the Moment Problem},
  author={Plaumann, Daniel},
  address={Konstanz},
  school={Universität Konstanz}
}

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Examination date of dissertation

April 28, 2008

Bibliography of Konstanz

Yes