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Some examples of quantifier elimination and o-minimality
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A structure with a total order that is dense without end-points is o-minimal if every definable set in dimension 1is a finite union of intervals and points. This notion materialized from observations that many of the proprieties of semialgebraic sets were deduced from very simple axioms, the ones that now define o-minimal structures. Indeed, o-minimality establishes strong regularity results of the definable sets. In this way, o-minimality can be viewed as a candidate to “topologie mod´er´ee” mentioned by Grothendieck in his Esquisse d’un programme. In the context of this dissertation, despite of its intrinsic richness, we study the property of quantifier elimination (abbreviated QE) as a way of proving o-minimality of a given structure. The goal of this dissertation was to study proofs of o-minimality and QE by studying a concrete example, the real closed ordered fields (abbreviated rcof). In Chapter 1, we begin by defining basic notions of first-order logic. We present some examples that will be useful later, such as the theory of rcof. We alude to the usefulness of different axiomatizations, such as the universal axiomatization, and simplifications of formulas, such as QE, that make the theories much more easier to understand. We present some criterias for a theory to admit QE. We present a geometrical perspective of the definable sets in general and the special case of o-minimality. In Chapter 2 we prove that the theory of rcof has QE. As a consequence we prove that every rcof is o-minimal. In Chapter 3 we study proprieties of o-minimal structures. In Chapter 4 we study the theory Tan of rcof with restricted analytic functions. We state that Tan has QE in the language Lan(1) and as a consequence we show that Tan admits an universal axiomatization in the language Lan(1; ( np)n=2;3;:::). In Chapter 5 we establish the result that every model of Tan can be seen as a substructure of a power series field R((t)). We use this fact to deduce key results concerning valuations on these structures and use these to prove that Tan(exp) admits QE and universal axiomatization, both in the language Lan(exp; log). In the last section of this chapter we begin by noting that, provided the theory admits QE, o-minimality is equivalent to regularity of the signal (whether it is greater, less or equal to zero) “at infinity” of the definable functions in one variable. This leads us to consider Hardy fields and using properties from these fields we prove that Tan(exp) is o-minimal.
Descrição
Tese de mestrado, Matemática, Universidade de Lisboa, Faculdade de ciências, 2021
Palavras-chave
o-minimalidade eliminação de quantificadores axiomatização universal corpos reais fechados ordenados corpos analíticos restritos Teses de mestrado - 2021