http://www.math.uni-konstanz.de/~schweigh/ Lecture notes Real Algebraic Geometry, Positivity and Convexity Academic Year 2016/2017 Markus Schweighofer Version of Thursday 30th August, 2018, 22:11 Universität Konstanz, Germany Preface. Chapters 1–4 are lecture notes of my course “Real Algebraic Geometry I” from the winter term 2016/2017. Chapters 5–8 are lecture notes of its continuation “Real Algebraic Geometry II” from the summer term 2017. Chapter 9 has been taught in my course “Geometry of Linear Matrix Inequalities” from the same summer term. Please report any ambiguities and errors (including typos) to: [email protected] This document is to a large extent based on the work of other people. For the relevant scientific sources, we refer to the literature referenced at the end of this document as well as the bibliographies of the books [ABR, BCR, BPR,KS, Mar,PD]. I would like to thank the numerous people that helped to improve these lecture notes: First of all, I thank Tom-Lukas Kriel, especially for coauthoring Chapter 9. Thanks go also to Sebas- tian Gruler and María López Quijorna and to those participants that pointed out errors and typos (among them I mention especially Alexander Taveira Blomenhofer, Nicolas Daans, Carl Eggen, Rüdiger Grunwald and Emre Öztürk in alphabetical order). Contents Introduction v 1 Ordered fields1 1.1 Orders of fields.................................1 1.2 Preorders.....................................7 1.3 Extensions of orders.............................. 10 1.4 Real closed fields................................ 13 1.5 Descartes’ rule of signs............................. 18 1.6 Counting real zeros with Hermite’s method................. 22 1.7 The real closure................................. 28 1.8 Real quantifier elimination........................... 31 1.9 Canonical isomorphisms of Boolean algebras of semialgebraic sets and classes...................................... 40 2 Hilbert’s 17th problem 43 2.1 Nonnegative polynomials in one variable.................. 43 2.2 Homogenization and dehomogenization.................. 45 2.3 Nonnegative quadratic polynomials..................... 47 2.4 The Newton polytope............................. 48 2.5 Artin’s solution to Hilbert’s 17th problem.................. 53 2.6 The Gram matrix method........................... 54 3 Prime cones and real Stellensätze 57 3.1 The real spectrum of a commutative ring.................. 57 3.2 Preorders and maximal prime cones..................... 62 3.3 Quotients and localization........................... 63 3.4 Abstract real Stellensätze............................ 64 3.5 The real radical ideal.............................. 66 3.6 Constructible sets................................ 67 3.7 Real Stellensätze................................. 70 4 Schmüdgen’s Positivstellensatz 75 4.1 The abstract Archimedean Positivstellensatz................ 75 4.2 The Archimedean Positivstellensatz [! §3.7]................ 76 4.3 Schmüdgen’s characterization of Archimedean preorders of the polyno- mial ring..................................... 77 iii iv 5 The real spectrum as a topological space 81 5.1 Tikhonov’s theorem............................... 81 5.2 Topologies on the real spectrum........................ 85 5.3 The real spectrum of polynomial rings.................... 88 5.4 The finiteness theorem for semialgebraic classes.............. 92 6 Semialgebraic geometry 97 6.1 Semialgebraic sets and functions....................... 97 6.2 The Łojasiewicz inequality........................... 101 6.3 The finiteness theorem for semialgebraic sets................ 104 7 Convex sets in vector spaces 109 7.1 The isolation theorem for cones........................ 109 7.2 Separating convex sets in topological vector spaces............ 114 7.3 Convex sets in locally convex vector spaces................. 118 7.4 Convex sets in finite-dimensional vector spaces............... 123 7.5 Application to ternary quartics........................ 132 8 Nonnegative polynomials with zeros 141 8.1 Modules over semirings............................ 141 8.2 Pure states on rings and ideals........................ 143 8.3 Dichotomy of pure states on ideals...................... 149 8.4 A local-global-principle............................ 152 9 Nonnegative polynomials and truncated quadratic modules 155 9.1 Pure states and polynomials over real closed fields............. 155 9.2 Degree bounds and quadratic modules................... 162 9.3 Concavity and Lagrange multipliers..................... 165 9.4 Linear polynomials and truncated quadratic modules........... 172 Tentative Lecture Notes Introduction The study of polynomial equations is a canonical subject in mathematics education, as is illustrated by the following examples: Quadratic equations in one variable (high school), systems of linear equations (linear algebra), polynomial equations in one vari- able and their symmetries (algebra, Galois theory), diophantine equations (number the- ory) and systems of polynomial equations (algebraic geometry, commutative algebra). In contrast to this, the study of polynomial inequalities (in the sense of “greater than” or “greater or equal than”) is mostly neglected even though it is much more important for applications: Indeed, in applications one often searches for a real solution rather than a complex one (as in classical algebraic geometry) and this solution must not necessarily be exact but only approximate. In a course about linear algebra there is frequently no time for linear optimization. An introductory course about algebra usually treats groups, rings and fields but disregards ordered and real closed fields as well as preorders or prime cones of rings. In a first course on algebraic geometry there is often no special attention paid to the real part of a variety and in commutative algebra quadratic modules are practically never treated. Most algebraists do not even know the notion of a preorder although it is as important for the study of systems of polynomial inequalities as the notion of an ideal is for the study of systems of polynomial equations. People from more applied areas such as numerical analysis, mathematical optimization or functional analysis know often more about real algebraic geometry than some algebraists, but often do not even recognize that polynomials play a decisive role in what they are doing. There are for example countless articles from functional analysis which are full of equations with binomial coefficients which turn out to be just disguised simple polynomial identities. In the same way as the study of polynomial systems of equations leads to the study of rings and their generalizations (such as modules), the study of systems of polynomial inequalities leads to the study of rings which are endowed with something that re- sembles an order. This additional structure raises many new questions that have to be clarified. These questions arise already at a very basic level so that we need as prereq- uisites only basic linear algebra, algebra and analysis. In particular, this course is really extremely well suited to students heading for a teaching degree. It includes several topics which are directly relevant for high school teaching. To arouse the reader’s curiosity, we present the following table. It contains on the left column notions we assume the reader is familiar with. On the right column we name v vi what could be seen more or less as their real counterparts mostly introduced in this course. Algebra Real Algebra Algebraic Geometry Real Algebraic Geometry systems of polynomial equations systems of polynomial inequalities “=” “≥” complex solutions real solutions C R algebraically closed fields real closed fields fields ordered fields ideals preorders prime ideals prime cones spectrum real spectrum Noetherian quasi-compact radical real radical fundamental theorem of algebra fundamental theorem of algebra Aachen, Aalborg, Aarhus, . Dortmund, Dublin, Innsbruck, . , Zagreb, Zürich . , Konstanz, Ljubljana, Rennes It is intended that the fundamental theorem of algebra appears on both sides of the table. In its usual form, it says that each non-constant univariate complex polynomial has a complex root. In Section 1.4, we will formulate it in a “real” way. The difficulties one has to deal with in the “real world” become already apparent when one asks the corresponding “real question”: When does a univariate complex polynomial have a real root? The answer to this will be given in Section 1.6 and requires already quite some thoughts. Traditionally, Real Algebraic Geometry has many ties with fields like Model Theory, Valuation Theory, Quadratic Form Theory and Algebraic Topology. In this lecture, we mainly emphasize however connections to fields like Optimization, Functional Analy- sis and Convexity that came up during the recent years and are now fully established. Throughout the lecture, N := f1, 2, 3, . g and N0 := f0g [ N denote the set of positive and nonnegative integers, respectively. Tentative Lecture Notes §1 Ordered fields 1.1 Orders of fields Reminder 1.1.1. Let M be a set. An order on M is a relation ≤ on M such that for all a, b, c 2 M: a ≤ a (reflexivity) (a ≤ b & b ≤ c) =) a ≤ c (transitivity) (a ≤ b & b ≤ a) =) a = b (antisymmetry) and a ≤ b or b ≤ a (linearity) In this case, (M, ≤) (or simply M if ≤ is clear from the context) is called an ordered set. For a, b 2 M, one defines a < b : () a ≤ b & a 6= b, a ≥ b : () b ≤ a and so on. Definition 1.1.2. Let (M, ≤1) and (N, ≤2) be ordered sets and j : M ! N be a map. Then j is called a homomorphism (of ordered sets) or monotonic if a ≤1 b =) j(a) ≤2 j(b) injective for all a, b 2 M. If j is and if bijective a ≤1 b () j(a) ≤2 j(b) embedding for all a, b 2 M, then j is called an (of ordered sets). isomorphism Proposition 1.1.3. Let (M, ≤1) and (N, ≤2) be ordered sets and j : M ! N a homomor- phism. Then the following are equivalent: (a) j is an embedding (b) j is injective (c) 8a, b 2 M : (j(a) ≤2 j(b) =) a ≤1 b) 1 2 Proof. (c) =) (b) Suppose (c) holds and let a, b 2 M such that j(a) = j(b). Then j(a) ≤2 j(b) and j(a) ≥2 j(b). Now (c) implies a ≤1 b and a ≥1 b. Hence a = b. (b) =) (c) Suppose (b) holds and let a, b 2 M with a 6≤1 b.