Positive Polynomials And Sum Of Squares In Formal Power Series Rings

A real field is a field where-1 is not a sum of squares.And a real closed field is a real field which has no proper real algebraic extensions.Main examples of real fields and real closed fields are the field of real numbers and fields of real algebraic numbers.In this paper we denote by R a real closed field.One of the main differences between a real field and a field is the existence of an ordering:it is well-known that a real field always admits at least one ordering.A positive polynomial in the ring of formal power series over a real field is a formal power series which is positive at all the orderings of the fraction field of this ring of formal power series.This thesis for Master Degree mainly studies the question that whether a positive polynomial is a sum of squares in the ring of formal power series over a real closed field.This question is similar to the 17th question of Hilbert.We need to consider this question via n,the number of variables.When n? 3,we give counter-examples to show that this question has a negative answer in general.When n=1 or 2,more advanced tools must be used,such as Baer-Krull theorem,and finally we prove that the question has a positive answer.