KOPS - The Institutional Repository of the University of Konstanz

Iterated rings of bounded elements and generalizations of Schmüdgen's theorem

Iterated rings of bounded elements and generalizations of Schmüdgen's theorem

Cite This

Files in this item

Checksum: MD5:21fba02d9dfa64d1657c7162dfae22a1

SCHWEIGHOFER, Markus, 2001. Iterated rings of bounded elements and generalizations of Schmüdgen's theorem [Dissertation]. Konstanz: University of Konstanz

@phdthesis{Schweighofer2001Itera-539, title={Iterated rings of bounded elements and generalizations of Schmüdgen's theorem}, year={2001}, author={Schweighofer, Markus}, address={Konstanz}, school={Universität Konstanz} }

terms-of-use Iterierte Ringe beschränkter Elemente und Verallgemeinerungen des Satzes von Schmüdgen Iterated rings of bounded elements and generalizations of Schmüdgen's theorem 2011-03-22T17:44:58Z Schweighofer, Markus eng 2011-03-22T17:44:58Z 2001 Schweighofer, Markus 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. application/pdf

Downloads since Oct 1, 2014 (Information about access statistics)

thesis.pdf 200

This item appears in the following Collection(s)

Search KOPS


Browse

My Account