Browsing by Author "Uotila, Valter Johan Edvard"
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Uotila, Valter Johan Edvard (2022)In this work, I prove the theorem of Bröcker and Scheiderer for basic open semialgebraic sets. The theorem provides an upper bound for a stability index of a real variety. The theory is based on real closed fields which generalize real numbers. A real variety is a subset of a real closed field that is defined by polynomial equalities. Every semialgebraic set is defined by a boolean combination of polynomial equations and inequalities of the sign conditions involving a finite number of polynomials. The basic semialgebraic sets are those semialgebraic sets that are defined solely by the sign conditions. In other words, we can construct semialgebraic sets from the basic semialgebraic sets by taking the finite unions, intersections, and complements of the basic semialgebraic sets. Then the stability index of a real variety indicates the upper bound of numbers of polynomials that are required to express an arbitrary semialgebraic subset of the variety. The theorem of Bröcker and Scheiderer shows that such upper bound exists and is finite for basic open semialgebraic subsets of a real variety. This work aims to be detailed in the proofs and represent sufficient prerequisites and references. The first chapter introduces the topic generally and motivates to study the theorem. The second chapter provides advanced prerequisites in algebra. One of such results is the factorial theorem of a total ring of fractions. Other advanced topics include radicals, prime ideals, associative algebras, a dimension of a ring, and various quotient structures. The third chapter defines real closed fields and semialgebraic sets that are the fundamental building blocks of the theory. The third chapter also develops the theory of quadratic forms. The main result of this chapter is Witt’s cancellation theorem. We also shortly describe the TsenLang theorem. The fourth chapter is about Pfister forms. Pfister forms are special kinds of quadratic forms that we extensively use in the proof of the main theorem. First, we define general Pfister forms over fields. Then we develop their theory over the fields of rational functions. Generally, Pfister forms share multiple similar properties as quadratic forms. The fifth chapter represents one and twodimensional examples of the main theorem. These examples are based on research that is done on constructive approaches to the theorem of Bröcker and Scheiderer. The examples clarify and motivate the result from an algorithmic perspective. Finally, we prove the main theorem of the work. The proof is heavily based on Pfister forms.
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