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Positivity is one of the most basic mathematical concepts. In many areas of mathematics (like analysis, real algebraic geometry, functional analysis, etc.) it shows up as positivity of a polynomial on a certain subset of R^n which itself is often given by polynomial inequalities. The main objective of the book is to give useful characterizations of such polynomials. It takes as starting point Hilbert's 17th Problem from 1900 and explains how E. Artin's solution of that problem eventually led to the development of real algebra towards the end of the 20th century. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed. Thus the monograph can also serve as the basis for a 2-semester course in real algebra.

Polynomials. --- Semialgebraic sets. --- Topological fields. --- 512.62 --- Fields. Polynomials --- 512.62 Fields. Polynomials --- Polynomials --- Semialgebraic sets --- Topological fields --- Algebra --- Algebraic fields --- Geometry, Algebraic --- Set theory --- Algebra. --- Algebraic geometry. --- Functional analysis. --- Algebraic Geometry. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Algebraic geometry --- Geometry --- Mathematics --- Mathematical analysis

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Mathematical Logic and Model Theory: A Brief Introduction offers a streamlined yet easy-to-read introduction to mathematical logic and basic model theory. It presents, in a self-contained manner, the essential aspects of model theory needed to understand model theoretic algebra. As a profound application of model theory in algebra, the last part of this book develops a complete proof of Ax and Kochen's work on Artin's conjecture about Diophantine properties of p-adic number fields. The character of model theoretic constructions and results differs significantly from that commonly found in algebra, by the treatment of formulae as mathematical objects. It is therefore indispensable to first become familiar with the problems and methods of mathematical logic.  Therefore, the text is divided into three parts: an introduction into mathematical logic (Chapter 1), model theory (Chapters 2 and 3), and the model theoretic treatment of several algebraic theories (Chapter 4). This book will be of interest to both advanced undergraduate and graduate students studying model theory and its applications to algebra. It may also be used for self-study.

Mathematical logic --- Mathematics --- wiskunde