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Sums of Squares in Local Rings

Ahmadieh, Arman | 2010

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 41134 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Gholamzadeh Mahmoudi, Mohammad
  7. Abstract:
  8. In this thesis, which is based on an article of Claus Scheiderer (Reference [21]), the relation between positive semi-definite elements of a semilocal ring, called psd, and elements of the ring which can be written as a sum of square elements of the ring and are called sos will be evaluated. In Chapter One, the concepts of prime cones, real spectrum of a ring and real ideals will be defined and then the relation between psd and sos elements in the total rings of quotients will be assessed. The Krull Valuation of an element which is a sum of squares will also be determined. In Chapter Two, the concept of totally positive elements in a ring will be defined and their properties will be studied. The relation between sos elements of a ring and its completion will be also studied in this chapter. In Chapter Three, the rings are of Krull Dimension one. In this chapter, the relation between psd and sos ele-ments will be evaluated in reduced rings. Then, the concepts of Japanese and Na-gata Rings will be introduced and the relation between psd and sos elements will be examined in Nagata Rings. Then, the relation between these elements will be analyzed in Formal Power Series Rings. Finally, a fundamental theorem will be stated in order to determine whether every psd element of a ring is sos or not
  9. Keywords:
  10. Valuation Ring ; Prime Cone ; Positive Semidefinite Solutions ; Sum of Squares (SOS) ; Local Ring

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