Iterated rings of bounded elements and generalizations of Schmüdgen's theorem
| dc.contributor.author | Schweighofer, Markus | |
| dc.date.accessioned | 2011-03-22T17:44:58Z | deu |
| dc.date.available | 2011-03-22T17:44:58Z | deu |
| dc.date.issued | 2001 | deu |
| dc.description.abstract | We consider a commutative algebra over the reals of finite transcendence degree. We call an element of it (geometrically) bounded if its square is bounded by a natural number on the whole real spectrum. We call it arithmetically bounded if the distance to the bound can even be described by a sum of squares of elements. In 1991, Schmüdgen proved in the case in which the algebra is finitely generated: If every element is geometrically bounded, then every element is even arithmetically bounded. This implies Schmüdgen's well-known Positivstellensatz which is used in optimization. In 1996, Becker and Powers considered the decreasing chain of iterated rings of bounded elements and showed that it becomes stable at the latest after the iteration given by the transcendence degree. In 1998, Monnier related both results and conjectured that this stable object contains exactly the arithmetically bounded elements. We prove this conjecture. An important application is the following generalization of Schmüdgen's Positivstellensatz: If an element is 'small at infinity' and nonnegative, then it becomes a sum of squares after adding an arbitrary small positive real number. | eng |
| dc.description.version | published | |
| dc.format.mimetype | application/pdf | deu |
| dc.identifier.ppn | 099210312 | deu |
| dc.identifier.uri | http://kops.uni-konstanz.de/handle/123456789/539 | |
| dc.language.iso | eng | deu |
| dc.legacy.dateIssued | 2002 | deu |
| dc.rights | terms-of-use | deu |
| dc.rights.uri | https://rightsstatements.org/page/InC/1.0/ | deu |
| dc.subject | Hilbertsches Problem 17 | deu |
| dc.subject | Positivstellensatz | deu |
| dc.subject | Satz von Schmüdgen | deu |
| dc.subject | Hilbert's 17th problem | deu |
| dc.subject | Positivstellensatz | deu |
| dc.subject | Schmüdgen's theorem | deu |
| dc.subject.ddc | 510 | deu |
| dc.subject.gnd | Mathematik | deu |
| dc.subject.gnd | Algebra | deu |
| dc.subject.gnd | Reelle Algebra | deu |
| dc.subject.gnd | Ring | deu |
| dc.subject.gnd | Holomorphiering | deu |
| dc.subject.gnd | Positives Polynom | deu |
| dc.subject.gnd | Momentenproblem | deu |
| dc.subject.msc | 13J30 | deu |
| dc.subject.msc | 44A60 | deu |
| dc.subject.msc | 11E25 | deu |
| dc.subject.msc | 14P10 | deu |
| dc.title | Iterated rings of bounded elements and generalizations of Schmüdgen's theorem | eng |
| dc.title.alternative | Iterierte Ringe beschränkter Elemente und Verallgemeinerungen des Satzes von Schmüdgen | deu |
| dc.type | DOCTORAL_THESIS | deu |
| dspace.entity.type | Publication | |
| kops.citation.bibtex | @phdthesis{Schweighofer2001Itera-539,
year={2001},
title={Iterated rings of bounded elements and generalizations of Schmüdgen's theorem},
author={Schweighofer, Markus},
address={Konstanz},
school={Universität Konstanz}
} | |
| kops.citation.iso690 | SCHWEIGHOFER, Markus, 2001. Iterated rings of bounded elements and generalizations of Schmüdgen's theorem [Dissertation]. Konstanz: University of Konstanz | deu |
| kops.citation.iso690 | SCHWEIGHOFER, Markus, 2001. Iterated rings of bounded elements and generalizations of Schmüdgen's theorem [Dissertation]. Konstanz: University of Konstanz | eng |
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<dcterms:abstract xml:lang="eng">We consider a commutative algebra over the reals<br />of finite transcendence degree.<br /><br /><br /><br />We call an element of it (geometrically) bounded<br />if its square is bounded by a natural number on<br />the whole real spectrum. We call it arithmetically<br />bounded if the distance to the bound can even be<br />described by a sum of squares of elements.<br /><br /><br /><br />In 1991, Schmüdgen proved in the case in which the<br />algebra is finitely generated: If every element<br />is geometrically bounded, then every element is<br />even arithmetically bounded. This implies<br />Schmüdgen's well-known Positivstellensatz which<br />is used in optimization.<br /><br /><br /><br />In 1996, Becker and Powers considered the<br />decreasing chain of iterated rings of bounded<br />elements and showed that it becomes stable<br />at the latest after the iteration given by the<br />transcendence degree.<br /><br /><br /><br />In 1998, Monnier related both results and<br />conjectured that this stable object contains<br />exactly the arithmetically bounded elements. We<br />prove this conjecture. An important application<br />is the following generalization of Schmüdgen's<br />Positivstellensatz: If an element is 'small at<br />infinity' and nonnegative, then it becomes a sum<br />of squares after adding an arbitrary small<br />positive real number.</dcterms:abstract>
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<dc:language>eng</dc:language>
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| kops.date.examination | 2002-04-15 | deu |
| kops.description.abstract | Wir betrachten eine kommutative Algebra über den<br />reellen Zahlen von endlichem Transzendenzgrad.<br /><br /><br /><br />Ein Element davon nennen wir (geometrisch)<br />beschränkt, wenn sein Quadrat auf dem gesamten<br />reellen Spektrum durch eine natürliche Zahl nach<br />oben beschränkt ist. Wir nennen es arithmetisch<br />beschränkt, wenn der Abstand zur Schranke sogar<br />durch eine Summe von Quadraten von Elementen<br />beschrieben werden kann.<br /><br /><br /><br />1991 bewies Schmüdgen für den Fall, daß die<br />Algebra endlich erzeugt ist: Wenn jedes Element<br />geometrisch beschränkt ist, dann ist jedes<br />Element sogar arithmetisch beschränkt. Daraus<br />folgt dann Schmüdgens bekannter<br />Positivstellensatz, der in der Optimierung<br />angewendet wird.<br /><br /><br /><br />1996 betrachteten Becker und Powers die<br />absteigende Kette iterierter Ringe von<br />beschränkten Elementen und bewiesen, daß diese<br />spätestens ab der durch den Transzendenzgrad<br />gegebenen Iteration stationär wird.<br /><br /><br /><br />1998 brachte Monnier diese beiden Ergebnisse in<br />Zusammenhang und vermutete, daß dieses stationäre<br />Objekt genau die arithmetisch beschränkten<br />Elemente enthält. Wir beweisen diese Vermutung.<br />Eine wichtige Anwendung ist dann folgende<br />Verallgemeinerung von Schmüdgens<br />Positivstellensatz: Falls ein Element 'klein im<br />Unendlichen' und nichtnegativ ist, dann ist es<br />nach Addition einer noch so kleinen positiven<br />reellen Zahl eine Summe von Quadraten. | deu |
| kops.description.openAccess | openaccessgreen | |
| kops.flag.knbibliography | false | |
| kops.identifier.nbn | urn:nbn:de:bsz:352-opus-8255 | deu |
| kops.opus.id | 825 | deu |
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