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

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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

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