Semialgebraic Geometry School of Mathematics and Statistics, University of Sydney A thesis submitted in fulfillment of the requirements for the degree of Master of Philosophy Mark S. Perrin 2020 1 Abstract In this thesis we introduce the preliminary results required to appreciate some key results of classical semialgebraic geometry. Namely, we give a detailed account of Sturm's root counting method, and Hermite's root counting method - both are used to count the number of real solutions of a finite system of polynomial equations in a single variable. After building a foothold in both the theoretical and algorithmic setting of the root counting methods, we extend to systems of polynomial equations and inequations, wherein lies the bridge to the definition of semialgebraic sets as a natural extension to algebraic sets. The main results covered are the cylindrical algebraic decomposition and triangulation of semial- gebraic sets, as well as Hardt's semialgebraic triviality of semialgebraic sets, where we show numerous consequences of each of these. We give the proofs of important concepts with a focus on intuitive exemplification and illustration. In the final section we discuss some improvements to the implementation of the standard conditions of Thom's lemma, 3.4.1, which may have implications on the efficiency of the projection operation used in cylindrical algebraic decomposition. 2 Acknowledgments I want to express my gratitude toward Laurentiu for his persistence and patience as my supervisor throughout my degree and writing of this thesis. I appreciate his diligence as a mentor - leading me with suggestions to seemingly independent personal discoveries and first-hand revelations in new topics in mathematics, and teaching me to always ask questions - which has undoubtedly improved my ability to learn. I also want to thank my family for the support relating to my studies or otherwise. They have always encouraged my pursuits in every way they can, with full faith that I will try my best to succeed, or at least make the most of the cards that might be dealt. Finally, I want to thank Gwen for being so supportive during our time to- gether including my studies, for being understanding in personal times, and for giving me motivation when it is most needed. 3 Contents 1 Counting real roots of polynomials 7 1.1 Sturm's method . .7 1.2 Tarski-Seidenberg - Elimination of a variable . 30 1.3 Hermite's method . 37 2 Semialgebraic sets 39 2.1 Semialgebraic sets - Definitions and examples . 39 2.2 Tarski-Seidenberg - Second form . 41 2.3 Tarski-Seidenberg - Third form . 42 2.4 Semialgebraic functions . 43 3 Decomposing semialgebraic sets 54 3.1 Cylindrical algebraic decomposition . 54 3.2 Constructing a c.a.d. adapted to a finite family of polynomials . 59 3.3 Algorithmic construction of an adapted c.a.d. 78 3.4 Improved cylindrical algebraic decomposition . 79 3.5 Dimension of semialgebraic sets . 101 3.6 Triangulation of semialgebraic sets . 110 4 Hardt's trivialization and consequences 123 4.1 Semialgebraic triviality of semialgebraic sets . 123 4.2 Semialgebraic Sard's theorem and upper bounds on connected components . 136 4.3 Algebraic computation trees and lower bounds on connected com- ponents . 148 5 Implementing the (?) condition 154 5.1 Checking the (?) condition . 154 5.2 Completing a family to satisfy the (?) condition . 161 A Hermite's Method 166 4 Introduction The goal of this thesis is to give a detailed discussion of several key results in semialgebraic geometry, emerging in the 1970's and onward. Of course, much of the underlying theory is older, relying on knowledge of algebraic geometry and analysis. We aim to give the new reader an introduction to the topic, providing rigorous and detailed proofs, complemented by an incremental building of the relevant theory, and illustrated with many worked examples. Following each of the main results (Theorems 3.4.7, 3.4.9, 3.6.1, and 4.1.1), we outline various consequences concerning the dimension, and the number of connected compo- nents of semialgebraic sets. Many of the results (and their proofs) we cover are taken from a preprint of Michel Coste's `Introduction to Semialgebraic Geom- etry' (2002), [1]. The structure of the paper reflects the logical dependence of the results, and hence the order in which the material should be learned. In Section 1 the root counting methods of Sturm and Hermite are introduced, providing a way to count the number of real solutions of a system of finitely many polynomial equations in a single variable. Sturm's method is developed further to count the number of solutions of a system consisting of a polynomial equation and several polynomial inequalities, and Hermite's method is devel- oped to relate the principal subresultant coefficients of two polynomials with the multiplicities of their common factors. This naturally gives a precursor for the definition of semialgebraic sets, as well as providing some powerful tools for their detailed examination. However, the reader may start with semialgebraic sets in Section 2, and return to the referenced material from Section 1 when necessary. The basic properties of semialgebraic sets are introduced in Section 2, particularly stability properties, as well as the tools commonly used to study semialgebraic sets. Semialgebraic sets are, by definition, sets of points in Rn satisfying finite boolean combinations of sign conditions on a family of polyno- mials, and being able to count the solutions of such a system is crucial in the method of cylindrical algebraic decomposition of semialgebraic sets into simpler, well-arranged subsets we call strata, or slightly more generally, cells. The reader will become familiar with taking projections of semialgebraic sets, which is used extensively throughout Section 3 as one of the primary methods used to study semialgebraic sets, and is responsible for some of the main features, including the cylindrical arrangement of cells, in a constructed cylindrical algebraic de- composition. Decomposing semialgebraic sets in this way provides information on the topology and dimension of the sets, and allows for the proof that every semialgebraic set can be triangulated, (Theorem 3.6.1). In Section 4 we intro- duce the notion of semialgebraic triviality in order to present Hardt's theorem as the next main result. We use the triangulation theorem to prove Hardt's the- orem, 4.1.1, on the triviality of semialgebraic sets, which states that the image of a semialgebraic set S by a continuous semialgebraic mapping h can be repre- sented as a finite union of subsets, over which the mapping is semialgebraically trivial, and that this triviality can be refined to be compatible with a finite collection of semialgebraic subsets of S. Hardt's theorem has consequences on the structure, dimension, and number of topological types of semialgebraic sets, 5 and is used to prove the semialgebraic version of Sard's theorem. Throughout, we aim to give illustrations and geometric exemplification of the key concepts in understanding these results. Some of the propositions/corollaries stated and proven throughout this thesis are posed as exercises in [1] (Coste). 6 1 Counting real roots of polynomials In this section we explore two methods for counting the real roots of polyno- mial equations with 1 variable. Namely, we consider Sturm's method, and the method of Hermite. Sturm's method invokes a variation of the Euclidean divi- sion algorithm to produce a sequence of polynomials, from which we are able to infer the number of distinct real roots of a polynomial on an open (not nec- essarily bounded) interval of the real line. We show how to extend this method to count the number of distinct real roots of a system of polynomial equations and inequalities in a single variable, and how to do so algorithmically, which will be used in Section 3 as a foundation for decomposing semialgebraic sets in an effective way. Hermite's method uses Newton sums, and principal minors of a matrix related to the Newton sums, to count the distinct complex roots of a polynomial, and even the distinct real roots of a polynomial. We show how to modify the setup in order to compute the number of solution of a system consist- ing of a polynomial equation and a polynomial inequality in a single variable. This method invokes Galois theory, principal subresultant coefficients, and a theorem of Jacobi on symmetric bilinear forms. We also show how the princi- pal subresultant coefficients of pairs of polynomials relate to common roots of these polynomials, which is an important feature we use in a construction in Section 3. While they are related, (see [2]), the methods of Sturm and Hermite are quite different in their application. Sturm's method requires branching of computation trees based on the signs of certain polynomials in the coefficients we start with, while Hermite's method avoids branching altogether. 1.1 Sturm's method Consider two nonzero polynomials P; Q 2 R[X]. We construct a sequence of polynomials P0;P1;:::;Pk by taking P0 := P , P1 := Q, and for i > 0 we define Pi+1 as the negative of the remainder of the Euclidean division of Pi−1 by Pi. That is, Pi−1 = PiAi + Ri for some Ai;Ri 2 R[X], and we define Pi+1 := −Ri. From this definition we can write Pi−1 = PiAi − Pi+1, or Pi+1 = PiAi − Pi−1 equivalently. We now define the Sturm sequence of P and Q as (P0;P1;:::;Pk), where Pk is the last nonzero polynomial in the sequence. Note that Pk = ± gcd(P; Q), since the process for computing Pk is the Euclidean algorithm up to a sign change. Example 1.