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dc.contributor.advisorMarshall, Murrayen_US
dc.contributor.advisorKuhlmann, Salmaen_US
dc.creatorFan, Weien_US
dc.date.accessioned2006-12-18T16:46:24Zen_US
dc.date.accessioned2013-01-04T05:11:59Z
dc.date.available2006-12-19T08:00:00Zen_US
dc.date.available2013-01-04T05:11:59Z
dc.date.created2006-12en_US
dc.date.issued2006-12-19en_US
dc.date.submittedDecember 2006en_US
dc.identifier.urihttp://hdl.handle.net/10388/etd-12182006-164624en_US
dc.description.abstractPositivity of polynomials, as a key notion in real algebra, is one of the oldest topics. In a given context, some polynomials can be represented in a form that reveals their positivity immediately, like sums of squares. A large body of literature deals with the question which positive polynomials can be represented in such a way.The milestone in this development was Schm"udgen's solution of the moment problem for compact semi-algebraic sets. In 1991, Schm"udgen proved that if the associated basic closed semi-algebraic set $K_{S}$ is compact, then any polynomial which is strictly positive on $K_{S}$ is contained in the preordering $T_{S}$.Putinar considered a further question: when are `linear representations' possible? He provided the first step in answering this question himself in 1993. Putinar proved if the quadratic module $M_{S}$ is archimedean, any polynomial which is strictly positive on $K_{S}$ is contained in $M_{S}$, i.e., has a linear representation.In the present thesis, we concentrate on the linear representations in the one variable polynomial ring. We first investigate the relationship of the two conditions in Schm"udgen's Theorem and Putinar's Criterion: $K_{S}$ compact and $M_{S}$ archimedean. They are actually equivalent. We find another proof for this result and hereby we can improve Schm"udgen's Theorem in the one variable case.Secondly, we investigate the relationship of $M_{S}$ and $T_{S}$. We use elementary arguments to prove in the one variable case when $K_{S}$ is compact, they are equal.Thirdly, we present Scheiderer's Main Theorem with a detailed proof. Scheiderer established a local-global principle for the polynomials non-negative on $K_{S}$ to be contained in $M_{S}$ in 2003. This principle which we call Scheiderer's Main Theorem here extends Putinar's Criterion.Finally, we consider Scheiderer's Main Theorem in the one variable case, and give a simplified version of this theorem. We also apply this Simple Version of the Main Theorem to give some elementary proofs for existing results.en_US
dc.language.isoen_USen_US
dc.subjectLinear Representationen_US
dc.subjectNon-negative Polynomialsen_US
dc.titleNon-negative polynomials on compact semi-algebraic sets in one variable caseen_US
thesis.degree.departmentMathematics and Statisticsen_US
thesis.degree.disciplineMathematics and Statisticsen_US
thesis.degree.grantorUniversity of Saskatchewanen_US
thesis.degree.levelMastersen_US
thesis.degree.nameMaster of Science (M.Sc.)en_US
dc.type.materialtexten_US
dc.type.genreThesisen_US
dc.contributor.committeeMemberMcQuillan, Ianen_US
dc.contributor.committeeMemberMartin, John R.en_US
dc.contributor.committeeMemberBremner, Murray R.en_US


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