Factorization in Generalized Power Series
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Bounded Pregeometries and Pairs of Fields
BOUNDED PREGEOMETRIES AND PAIRS OF FIELDS ∗ LEONARDO ANGEL´ and LOU VAN DEN DRIES For Francisco Miraglia, on his 70th birthday Abstract A definable set in a pair (K, k) of algebraically closed fields is co-analyzable relative to the subfield k of the pair if and only if it is almost internal to k. To prove this and some related results for tame pairs of real closed fields we introduce a certain kind of “bounded” pregeometry for such pairs. 1 Introduction The dimension of an algebraic variety equals the transcendence degree of its function field over the field of constants. We can use transcendence degree in a similar way to assign to definable sets in algebraically closed fields a dimension with good properties such as definable dependence on parameters. Section 1 contains a general setting for model-theoretic pregeometries and proves basic facts about the dimension function associated to such a pregeometry, including definable dependence on parameters if the pregeometry is bounded. One motivation for this came from the following issue concerning pairs (K, k) where K is an algebraically closed field and k is a proper algebraically closed subfield; we refer to such a pair (K, k)asa pair of algebraically closed fields, and we consider it in the usual way as an LU -structure, where LU is the language L of rings augmented by a unary relation symbol U. Let (K, k) be a pair of algebraically closed fields, and let S Kn be definable in (K, k). Then we have several notions of S being controlled⊆ by k, namely: S is internal to k, S is almost internal to k, and S is co-analyzable relative to k. -
Formal Power Series - Wikipedia, the Free Encyclopedia
Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series -
Sturm's Theorem
Sturm's Theorem: determining the number of zeroes of polynomials in an open interval. Bachelor's thesis Eric Spreen University of Groningen [email protected] July 12, 2014 Supervisors: Prof. Dr. J. Top University of Groningen Dr. R. Dyer University of Groningen Abstract A review of the theory of polynomial rings and extension fields is presented, followed by an introduction on ordered, formally real, and real closed fields. This theory is then used to prove Sturm's Theorem, a classical result that enables one to find the number of roots of a polynomial that are contained within an open interval, simply by counting the number of sign changes in two sequences. This result can be extended to decide the existence of a root of a family of polynomials, by evaluating a set of polynomial equations, inequations and inequalities with integer coefficients. Contents 1 Introduction 2 2 Polynomials and Extensions 4 2.1 Polynomial rings . .4 2.2 Degree arithmetic . .6 2.3 Euclidean division algorithm . .6 2.3.1 Polynomial factors . .8 2.4 Field extensions . .9 2.4.1 Simple Field Extensions . 10 2.4.2 Dimensionality of an Extension . 12 2.4.3 Splitting Fields . 13 2.4.4 Galois Theory . 15 3 Real Closed Fields 17 3.1 Ordered and Formally Real Fields . 17 3.2 Real Closed Fields . 22 3.3 The Intermediate Value Theorem . 26 4 Sturm's Theorem 27 4.1 Variations in sign . 27 4.2 Systems of equations, inequations and inequalities . 32 4.3 Sturm's Theorem Parametrized . 33 4.3.1 Tarski's Principle . -
Chapter 10 Orderings and Valuations
Chapter 10 Orderings and valuations 10.1 Ordered fields and their natural valuations One of the main examples for group valuations was the natural valuation of an ordered abelian group. Let us upgrade ordered groups. A field K together with a relation < is called an ordered field and < is called an ordering of K if its additive group together with < is an ordered abelian group and the ordering is compatible with the multiplication: (OM) 0 < x ^ 0 < y =) 0 < xy . For the positive cone of an ordered field, the corresponding additional axiom is: (PC·)P · P ⊂ P . Since −P·−P = P·P ⊂ P, all squares of K and thus also all sums of squares are contained in P. Since −1 2 −P and P \ −P = f0g, it follows that −1 2= P and in particular, −1 is not a sum of squares. From this, we see that the characteristic of an ordered field must be zero (if it would be p > 0, then −1 would be the sum of p − 1 1's and hence a sum of squares). Since the correspondence between orderings and positive cones is bijective, we may identify the ordering with its positive cone. In this sense, XK will denote the set of all orderings resp. positive cones of K. Let us consider the natural valuation of the additive ordered group of the ordered field (K; <). Through the definition va+vb := vab, its value set vK becomes an ordered abelian group and v becomes a homomorphism from the multiplicative group of K onto vK. We have obtained the natural valuation of the ordered field (K; <). -
Chapter 1 the Field of Reals and Beyond
Chapter 1 The Field of Reals and Beyond Our goal with this section is to develop (review) the basic structure that character- izes the set of real numbers. Much of the material in the ¿rst section is a review of properties that were studied in MAT108 however, there are a few slight differ- ences in the de¿nitions for some of the terms. Rather than prove that we can get from the presentation given by the author of our MAT127A textbook to the previous set of properties, with one exception, we will base our discussion and derivations on the new set. As a general rule the de¿nitions offered in this set of Compan- ion Notes will be stated in symbolic form this is done to reinforce the language of mathematics and to give the statements in a form that clari¿es how one might prove satisfaction or lack of satisfaction of the properties. YOUR GLOSSARIES ALWAYS SHOULD CONTAIN THE (IN SYMBOLIC FORM) DEFINITION AS GIVEN IN OUR NOTES because that is the form that will be required for suc- cessful completion of literacy quizzes and exams where such statements may be requested. 1.1 Fields Recall the following DEFINITIONS: The Cartesian product of two sets A and B, denoted by A B,is a b : a + A F b + B . 1 2 CHAPTER 1. THE FIELD OF REALS AND BEYOND A function h from A into B is a subset of A B such that (i) 1a [a + A " 2bb + B F a b + h] i.e., dom h A,and (ii) 1a1b1c [a b + h F a c + h " b c] i.e., h is single-valued. -
Super-Real Fields—Totally Ordered Fields with Additional Structure, by H. Garth Dales and W. Hugh Woodin, Clarendon Press
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 35, Number 1, January 1998, Pages 91{98 S 0273-0979(98)00740-X Super-real fields|Totally ordered fields with additional structure, by H. Garth Dales and W. Hugh Woodin, Clarendon Press, Oxford, 1996, 357+xiii pp., $75.00, ISBN 0-19-853643-7 1. Automatic continuity The structure of the prime ideals in the ring C(X) of real-valued continuous functions on a topological space X has been studied intensively, notably in the early work of Kohls [6] and in the classic monograph by Gilman and Jerison [4]. One can come to this in any number of ways, but one of the more striking motiva- tions for this type of investigation comes from Kaplansky’s study of algebra norms on C(X), which he rephrased as the problem of “automatic continuity” of homo- morphisms; this will be recalled below. The present book is concerned with some new and very substantial ideas relating to the classification of these prime ideals and the associated rings, which bear on Kaplansky’s problem, though the precise connection of this work with Kaplansky’s problem remains largely at the level of conjecture, and as a result it may appeal more to those already in possession of some of the technology to make further progress – notably, set theorists and set theoretic topologists and perhaps a certain breed of model theorist – than to the audience of analysts toward whom it appears to be aimed. This book does not supplant either [4] or [2] (etc.); it follows up on them. -
Pseudo Real Closed Field, Pseudo P-Adically Closed Fields and NTP2
Pseudo real closed fields, pseudo p-adically closed fields and NTP2 Samaria Montenegro∗ Université Paris Diderot-Paris 7† Abstract The main result of this paper is a positive answer to the Conjecture 5.1 of [15] by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC field, then T h(M) is NTP2 if and only if M is bounded. In the case of PpC fields, we prove that if M is a bounded PpC field, then T h(M) is NTP2. We also generalize this result to obtain that, if M is a bounded PRC or PpC field with exactly n orders or p-adic valuations respectively, then T h(M) is strong of burden n. This also allows us to explicitly compute the burden of types, and to describe forking. Keywords: Model theory, ordered fields, p-adic valuation, real closed fields, p-adically closed fields, PRC, PpC, NIP, NTP2. Mathematics Subject Classification: Primary 03C45, 03C60; Secondary 03C64, 12L12. Contents 1 Introduction 2 2 Preliminaries on pseudo real closed fields 4 2.1 Orderedfields .................................... 5 2.2 Pseudorealclosedfields . .. .. .... 5 2.3 The theory of PRC fields with n orderings ..................... 6 arXiv:1411.7654v2 [math.LO] 27 Sep 2016 3 Bounded pseudo real closed fields 7 3.1 Density theorem for PRC bounded fields . ...... 8 3.1.1 Density theorem for one variable definable sets . ......... 9 3.1.2 Density theorem for several variable definable sets. ........... 12 3.2 Amalgamation theorems for PRC bounded fields . ........ 14 ∗[email protected]; present address: Universidad de los Andes †Partially supported by ValCoMo (ANR-13-BS01-0006) and the Universidad de Costa Rica. -
Properties of the Real Numbers Drexel University PGSA Informal Talks
Properties of the real numbers Drexel University PGSA Informal Talks Sam Kennerly January 2012 ★ We’ve used the real number line since elementary school, but that’s not the same as defining it. ★ We’ll review the synthetic approach with Hilbert’s definition of R as the complete ordered field. 1 Confession: I was an undergrad mathematics major. Why bother? ★ Physical theories often require explicit or implicit assumptions about real numbers. ★ Quantum mechanics assumes that physical measurement results are eigenvalues of a Hermitian operator, which must be real numbers. State vectors and wavefunctions can be complex. Are they physically “real”? wavefunctions and magnitudes ★ Special relativity can use imaginary time, which is related to Wick rotation in some quantum gravity theories. Is imaginary time unphysical? (SR can also be described without imaginary time. Misner, Thorne & Wheeler recommend never using ıct as a time coordinate.) ★ Is spacetime discrete or continuous? Questions like these require a rigorous definition of “real.” imaginary time? 2 Spin networks in loop quantum gravity assume discrete time evolution. Mini-biography of David Hilbert ★ First to use phrase “complete ordered field.” ★ Published Einstein’s GR equation before Einstein! (For details, look up Einstein-Hilbert action.) ★ Chairman of the Göttingen mathematics department, which had enormous influence on modern physics. (Born, Landau, Minkowski, Noether, von Neumann, Wigner, Weyl, et al.) ★ Ahead of his time re: racism, sexism, nationalism. “Mathematics knows no races or geographic boundaries.” “We are a university, not a bath house.” (in support of hiring Noether) David Hilbert in 1912 Minister Rust: "How is mathematics in Göttingen now that it has been freed of the Jewish influence?” Hilbert: “There is really none any more.” ★ Hilbert-style formalism (my paraphrasing): 1. -
Real Closed Fields
University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 1968 Real closed fields Yean-mei Wang Chou The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Chou, Yean-mei Wang, "Real closed fields" (1968). Graduate Student Theses, Dissertations, & Professional Papers. 8192. https://scholarworks.umt.edu/etd/8192 This Thesis is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. EEAL CLOSED FIELDS By Yean-mei Wang Chou B.A., National Taiwan University, l96l B.A., University of Oregon, 19^5 Presented in partial fulfillment of the requirements for the degree of Master of Arts UNIVERSITY OF MONTANA 1968 Approved by: Chairman, Board of Examiners raduate School Date Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: EP38993 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI OwMTtation PVblmhing UMI EP38993 Published by ProQuest LLC (2013). Copyright in the Dissertation held by the Author. -
Chapter I, the Real and Complex Number Systems
CHAPTER I THE REAL AND COMPLEX NUMBERS DEFINITION OF THE NUMBERS 1, i; AND p2 In order to make precise sense out of the concepts we study in mathematical analysis, we must first come to terms with what the \real numbers" are. Everything in mathematical analysis is based on these numbers, and their very definition and existence is quite deep. We will, in fact, not attempt to demonstrate (prove) the existence of the real numbers in the body of this text, but will content ourselves with a careful delineation of their properties, referring the interested reader to an appendix for the existence and uniqueness proofs. Although people may always have had an intuitive idea of what these real num- bers were, it was not until the nineteenth century that mathematically precise definitions were given. The history of how mathematicians came to realize the necessity for such precision in their definitions is fascinating from a philosophical point of view as much as from a mathematical one. However, we will not pursue the philosophical aspects of the subject in this book, but will be content to con- centrate our attention just on the mathematical facts. These precise definitions are quite complicated, but the powerful possibilities within mathematical analysis rely heavily on this precision, so we must pursue them. Toward our primary goals, we will in this chapter give definitions of the symbols (numbers) 1; i; and p2: − The main points of this chapter are the following: (1) The notions of least upper bound (supremum) and greatest lower bound (infimum) of a set of numbers, (2) The definition of the real numbers R; (3) the formula for the sum of a geometric progression (Theorem 1.9), (4) the Binomial Theorem (Theorem 1.10), and (5) the triangle inequality for complex numbers (Theorem 1.15). -
2 the Real Numbers As a Complete Ordered Field
Section 2: The Axioms for the Real Numbers 12 2 The real numbers as a complete ordered eld In this section are presented what can be thought of as “the rules of the game:” the axioms of the real numbers. In this work, we present these axioms as rules without justication. There are other approaches which can be used. For example, another standard technique is to begin with the Peano axioms—the axioms of the natural numbers—and build up to the real numbers through several “completions” of this system. In such a setup, our axioms are theorems. 2.1 Field Axioms This rst set of axioms are called the eld axioms because any object satisfying them is called a eld. They give the algebraic properties of the real numbers. A eld is a nonempty set F along with two functions, multiplication : F F → F and addition + : F F → F satisfying the following axioms.3 Axiom 1 (Associative Laws). If a, b, c ∈ F, then (a + b)+c=a+(b+c) and (a b) c = a (b c). Axiom 2 (Commutative Laws). If a, b ∈ F, then a + b = b + a and a b = b a. Axiom 3 (Distributive Law). If a, b, c ∈ F, then a (b + c)=ab+ac. Axiom 4 (Existence of identities). There are 0, 1 ∈ F such that a +0=a and a 1=a,∀a∈F. Axiom 5 (Existence of an additive inverse). For each a ∈ F there is a ∈ F such that a +(a)=0. Axiom 6 (Existence of a multiplicative inverse). For each a ∈ F \{0} there is a1 ∈ F such that a a1 =1. -
Introduction to the Real Spectrum
INTRODUCTION TO THE REAL SPECTRUM PETE L. CLARK 1. Introduction About eighteen months ago I was trying to learn a little bit about model theory for a student conference (the 2003 Arizona Winter School) explaining the connec- tions between model theory and arithmetic geometry. There are indeed deep and important connections here, and I was quickly convinced that model theory is an area of mathematics that it is good for everyone to know at least a little bit about, but that is not my point here. Rather, while reading a basic text, they cited as examples of basic algebraic struc- tures: group, ring, module, ¯eld, ordered ¯eld, . , and here I stopped: ordered ¯eld?!? Not since the tedious axiomatic description of R in my ¯rst undergraduate analysis class had I even encountered the term; what kind of loon would regard an ordered ¯eld as an object as basic as the others in the list? This talk (given in 12/04 at Colorado College) is a meditation on how wrong I was.1 After a brief review of the notion of an ordered ¯eld (it's what you think it is!), we will aim to engage { and to some extent, to answer { the following questions: ² Which ¯elds can be ordered? ² If a ¯eld can be ordered, in how many di®erent ways can it be ordered? ² What information about a ¯eld is encoded in the set of all of its orderings? ² Can we regard the set of orderings of a ¯eld as some kind of space? ² When the ¯eld is the ¯eld of functions of an algebraic variety over a real-closed ¯eld R, how is this space related to the set of R-points of the variety? 2.