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      Non-negative polynomials on compact semi-algebraic sets in one variable case

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      Date
      2006-12-19
      Author
      Fan, Wei
      Type
      Thesis
      Degree Level
      Masters
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      Abstract
      Positivity 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.
      Degree
      Master of Science (M.Sc.)
      Department
      Mathematics and Statistics
      Program
      Mathematics and Statistics
      Supervisor
      Marshall, Murray; Kuhlmann, Salma
      Committee
      McQuillan, Ian; Martin, John R.; Bremner, Murray R.
      Copyright Date
      December 2006
      URI
      http://hdl.handle.net/10388/etd-12182006-164624
      Subject
      Linear Representation
      Non-negative Polynomials
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