On Numbers, Germs, and Transseries
|
|
- Poppy Pearson
- 5 years ago
- Views:
Transcription
1 On Numbers, Germs, and Transseries Matthias Aschenbrenner UCLA Lou van den Dries UIUC Joris van der Hoeven CNRS
2 (surreal) Numbers Introduction 2/28 Three intimately related topics... Germs (in HARDY fields) Transseries
3 Introduction 2/28 Germs (in HARDY fields)
4 HARDY fields 3/28 Let C 1 be the ring of germs at + of continuously differentiable functions (a, ) R (a R). We denote the germ at + of a function f also by f, relying on context. Definition A HARDY field is a subring of C 1 which is a field that contains with each germ of a function f also the germ of its derivative f (where f might be defined on a smaller interval than f). Examples Q, R, R(x), R(x, e x ), R(x, e x, log x), R x, e x2, erf x
5 HARDY fields 4/28 HARDY fields capture the somewhat vague notion of functions with regular growth at infinity (BOREL, DU BOIS-REYMOND,...): Let H be a HARDY field and f H. Then f 0 1 f H { f (t)>0, eventually, or f (t) < 0, eventually. Consequently, H carries an ordering making H an ordered field: f is eventually monotonic, and f > 0 f (t)>0 eventually; lim f (t) R {± }. t +
6 Transseries 5/28 (surreal) Numbers Germs (in HARDY fields) Transseries
7 Transseries 5/28 Transseries
8 The field T of transseries 6/28 T closure of R {x} under exp, log and infinite summation e ex + 3 e x2 +5(log x) π +42 +x 1 + +e x Here one should think of x as a positive infinite indeterminate.
9 The field T of transseries 6/28 T closure of R {x} under exp, log and infinite summation e ex +e x/2 + 3 e x2 +5(log x) π +42 +x 1 +2x 2 + +e x Here one should think of x as a positive infinite indeterminate.
10 The field T of transseries 6/28 T closure of R {x} under exp, log and infinite summation e ex +e x/2 +e x/3 + 3 e x2 +5(log x) π +42 +x 1 +2x 2 +6x 3 + +e x Here one should think of x as a positive infinite indeterminate.
11 The field T of transseries 6/28 T closure of R {x} under exp, log and infinite summation e ex +e x/2 +e x/3 + 3 e x2 +5(log x) π +42 +x 1 +2x 2 +6x x 4 + +e x Here one should think of x as a positive infinite indeterminate.
12 The field T of transseries 6/28 T closure of R {x} under exp, log and infinite summation f m m = e ex +e x/2 + 3 e x2 +5(log x) π +42 +x 1 +2x 2 +6x 3 + +e x m x: positive infinite indeterminate f m : coefficent m: transmonomial
13 The field T of transseries 6/28 T closure of R {x} under exp, log and infinite summation m f m m = e ex +e x/2 + 3 e x2 +5(log x) π +42 +x 1 +2x 2 +6x 3 + +e x x: positive infinite indeterminate f m : coefficent m: transmonomial The formal definition of T is inductive and somewhat lengthy. For each transseries there is a finite bound on the nesting of exp and log among its transmonomials: the following transseries are not in T: 1 x + 1 e x e ex e eex 1 x + 1 x log x + 1 x log x log log x +
14 T as an ordered differential field 7/28 With the natural ordering of transseries (via the leading coefficient), T is a real closed ordered field extension of R. Each f T can be differentiated term by term (with x =1): ( n=0 n! ex x n) ) ) = n=0 n!( ex x n ) = n=0 n!( ex x n n ex ) x n+1 = ex x This yields a derivation f f on the field T: ( f + g) = f + g, ( f g) = f g + f g Its constant field is { f T: f =0} =R. Given f, g T, the equation y + fy= g admits a solution y 0 in T.
15 Surreal numbers 8/28 (surreal) Numbers Germs (in HARDY fields) Transseries
16 Surreal numbers 8/28 (surreal) Numbers
17 Surreal numbers 9/28 These are simply strings of +, of arbitrary ordinal length. CONWAY turned the class No of surreals into a real closed field extension of R.
18 Addition of surreal numbers 10/28 x No is simpler than y No : x is a prefix of y For subsets L<R of No, let {L R} be the simplest x No with L<x<R. Any x No is of the form x ={L R} for suitable subsets L<R of No. Example 0 ={ }, 1 ={0 }, 2 ={0, 1 }, 1 ={0 1}, ω ={0, 1, } 2 Definition If x ={x L x R } and y ={y L y R }, then x +y {x L +y, x +y L x R +y, x +y R }. (Idea: we want x L +y < x +y < x R +y, )
19 Exponentiation and differentiation 11/28 In the 1980s, GONSHOR (based on ideas of KRUSKAL) defined an exponential function exp: No No >0 that extends x e x on R. In 2006, BERARDUCCI and MANTOVA (using ideas of VDH and SCHMELING) defined a derivation BM on No with ker BM =R, BM (ω)=1, BM (exp( f ))= BM ( f ) exp( f ) for f No. In a certain technical sense, it is the simplest such derivation that satisfies some natural further conditions. The BM-derivation on No behaves in many ways like the derivation on T, with ω > R in the role of x > R. For instance, BM (log ω)= 1 ω.
20 Towards a unified theory 12/28 (surreal) Numbers Germs (in HARDY fields) Transseries
21 Towards a unified theory 12/28 (surreal) Numbers? Transseries Germs (in HARDY fields)
22 Towards a unified theory 12/28 (surreal) Numbers H-fields Transseries Germs (in HARDY fields)
23 Towards a unified theory 12/28 (surreal) Numbers Hardy Hausdorff Hahn H-fields Transseries Germs (in HARDY fields)
24 Asymptotic relations 13/28 Let K be an ordered differential field with constant field C ={ f K : f =0}. We define f g : f c g for some c C >0 ( f is dominated by g) f g : f c g for all c C >0 ( f is negligible w.r.t. g) f g : f g f ( f is asymptotic to g) f g : f g g ( f is equivalent to g) Example In T: 0 e x x log x x 1/10 e x e x +x e ex
25 Definition We call K an H-field if H1. f > C f >0; H2. f 1 f c for some c C. Examples H-fields 14/28 HARDY fields containing R; ordered differential subfields of T or No that contain R. T admits further elementary properties in addition to being an H-field. It has small derivation, that is, f 1 f 1; and is LIOUVILLE closed, that is, it is real closed and for all f, g, there is some y 0 with y + fy= g.
26 One of our main results 15/28 We view T model-theoretically as a structure with the primitives 0, 1, +,, (derivation), (ordering). Theorem (Ann. of Math. Stud. vol. 195) The elementary theory of T is completely axiomatized by: 1 2 T is a LIOUVILLE closed H-field with small derivation; T satisfies the intermediate value property for differential polynomials. Actually 2 is a bit of an afterthought. A corollary of the theorem: the theory of T is decidable. We also prove a quantifier elimination result for T in a natural expansion of the above language.
27 H-field elements as germs 16/28 (surreal) Numbers H-fields Transseries Germs (in HARDY fields)
28 H-field elements as germs 16/28 (surreal) Numbers Transseries Germs (in HARDY fields) H-fields
29 Closure properties of HARDY fields 17/28 Theorem (HARDY 1910, BOURBAKI 1951) Any HARDY field has a smallest LIOUVILLE closed HARDY field extension.
30 Closure properties of HARDY fields 17/28 Theorem (HARDY 1910, BOURBAKI 1951) Any HARDY field has a smallest LIOUVILLE closed HARDY field extension. Theorem (SINGER 1975) Let H be a HARDY field and Φ H(Y) be a rational function. If y C 1 satisfies the differential equation y =Φ(y), then H y =H(y, y ) is a HARDY field.
31 Closure properties of HARDY fields 17/28 Theorem (HARDY 1910, BOURBAKI 1951) Any HARDY field has a smallest LIOUVILLE closed HARDY field extension. Theorem (SINGER 1975) Let H be a HARDY field and Φ H(Y) be a rational function. If y C 1 satisfies the differential equation y =Φ(y), then H y =H(y, y ) is a HARDY field. Remark (BOSHERNITZAN 1987) Any solution y C 1 to y +y =e x2 lies in a HARDY field, but any HARDY field contains at most one solution.
32 Conjecture Conjectural closure properties 18/28 Let H be a maximal HARDY field. Then A B H satisfies the differential intermediate value property. For countable subsets A<B of H, there exists an h H with A<h<B.
33 Conjecture Conjectural closure properties 18/28 Let H be a maximal HARDY field. Then A B A B H satisfies the differential intermediate value property. For countable subsets A<B of H, there exists an h H with A<h<B. Corollary H is elementarily equivalent to T as an ordered differential field. Under CH, all maximal HARDY fields are isomorphic.
34 Work in progress on Conjecture A 19/28 Theorem (VAN DER HOEVEN 2009) The subfield T da of transseries that satisfy an algebraic differential equation over R can be embedded (as an ordered differential field) in a HARDY field.
35 Work in progress on Conjecture A 19/28 Theorem (VAN DER HOEVEN 2009) The subfield T da of transseries that satisfy an algebraic differential equation over R can be embedded (as an ordered differential field) in a HARDY field. y = e ex +y 2
36 Work in progress on Conjecture A 19/28 Theorem (VAN DER HOEVEN 2009) The subfield T da of transseries that satisfy an algebraic differential equation over R can be embedded (as an ordered differential field) in a HARDY field. y = (e ex +y 2 ) = e ex + e ex 2 +2 e e x 3 +
37 Work in progress on Conjecture A 19/28 Theorem (VAN DER HOEVEN 2009) The subfield T da of transseries that satisfy an algebraic differential equation over R can be embedded (as an ordered differential field) in a HARDY field. y = (e ex +y 2 ) = e ex + e ex 2 +2 e e x 3 + Theorem in progress Any HARDY field has an ω-free HARDY field extension. Any ω-free HARDY field has a newtonian differentially algebraic HARDY field extension.
38 Theorem (BOREL 1895) Work in progress on Conjecture B 20/28 Any y R[[x 1 ]] is the asymptotic expansion of a germ ỹ in C.
39 Theorem (BOREL 1895) Work in progress on Conjecture B 20/28 Any y R[[x 1 ]] is the asymptotic expansion of a germ ỹ in C. Example y =x 1 +2!!x 2 +3!!x 3 + is differentially transcendental over R the differential field R ỹ =R(ỹ, ỹ, ỹ, ) is a HARDY field.
40 Theorem (BOREL 1895) Work in progress on Conjecture B 20/28 Any y R[[x 1 ]] is the asymptotic expansion of a germ ỹ in C. Example y =x 1 +2!!x 2 +3!!x 3 + is differentially transcendental over R the differential field R ỹ =R(ỹ, ỹ, ỹ, ) is a HARDY field. Theorem in progress Every pseudocauchy sequence (y n ) in a HARDY field H has a pseudolimit in some HARDY field extension of H. Conjecture B for countable A and B = (SJÖDIN 1970). General case.
41 H-field elements as surreal numbers 21/28 (surreal) Numbers H-fields Transseries Germs (in HARDY fields)
42 H-field elements as surreal numbers 21/28 (surreal) Numbers H-fields Transseries Germs (in HARDY fields)
43 Embedding H-fields into the surreals 22/28 Theorem (to appear in JEMS) Every H-field with small derivation and constant field R can be embedded as an ordered differential field into No.
44 Embedding H-fields into the surreals 22/28 Theorem (to appear in JEMS) Every H-field with small derivation and constant field R can be embedded as an ordered differential field into No. Theorem (to appear in JEMS) Let κ be an uncountable cardinal. The field No(κ) of surreal numbers of length <κ is an elementary submodel of No.
45 Embedding H-fields into the surreals 22/28 Theorem (to appear in JEMS) Every H-field with small derivation and constant field R can be embedded as an ordered differential field into No. Theorem (to appear in JEMS) Let κ be an uncountable cardinal. The field No(κ) of surreal numbers of length <κ is an elementary submodel of No. Corollary in progress Under CH all maximal HARDY fields are isomorphic to No(ω 1 ).
46 H-field elements as transseries 23/28 (surreal) Numbers H-fields Transseries Germs (in HARDY fields)
47 H-field elements as transseries 23/28 (surreal) Numbers H-fields Transseries Germs (in HARDY fields)
48 H-field elements as transseries 23/28 Germs (in HARDY fields) H-fields (surreal) Numbers Transseries
49 How to generalize transseries 24/28 Definition (VAN DER HOEVEN 2000, SCHMELING 2001) A field T =R[[M]] with partial log: T T is a field of transseries if T1. The domain of log is T >0 ; T2. for each m M and n supp log m, we have n 1; T3. log (1 +ε)=ε 1 2 ε ε3 +, for all ε T with ε 1; T4. for every sequence (m n ) M N with m n+1 supp log m n for all n, there exists an index n 0 such that for all n>n 0 and all n supp log m n, we have n m n+1 and (log m n ) mn+1 =±1.
50 How to generalize transseries 24/28 Definition (VAN DER HOEVEN 2000, SCHMELING 2001) A field T =R[[M]] with partial log: T T is a field of transseries if T1. The domain of log is T >0 ; T2. for each m M and n supp log m, we have n 1; T3. log (1 +ε)=ε 1 2 ε ε3 +, for all ε T with ε 1; T4. for every sequence (m n ) M N with m n+1 supp log m n for all n, there exists an index n 0 such that for all n>n 0 and all n supp log m n, we have n m n+1 and (log m n ) mn+1 =±1. x +e logx+e loglogx+e x +e logx+e loglogx+e +loglogx +loglogx +log x
51 How to generalize transseries 24/28 Definition (VAN DER HOEVEN 2000, SCHMELING 2001) A field T =R[[M]] with partial log: T T is a field of transseries if T1. The domain of log is T >0 ; T2. for each m M and n supp log m, we have n 1; T3. log (1 +ε)=ε 1 2 ε ε3 +, for all ε T with ε 1; T4. for every sequence (m n ) M N with m n+1 supp log m n for all n, there exists an index n 0 such that for all n>n 0 and all n supp log m n, we have n m n+1 and (log m n ) mn+1 =±1. x +e logx+e loglogx+e x +e logx+e loglogx+e +loglogx +loglogx +log x
52 How to generalize transseries 24/28 Definition (VAN DER HOEVEN 2000, SCHMELING 2001) A field T =R[[M]] with partial log: T T is a field of transseries if T1. The domain of log is T >0 ; T2. for each m M and n supp log m, we have n 1; T3. log (1 +ε)=ε 1 2 ε ε3 +, for all ε T with ε 1; T4. for every sequence (m n ) M N with m n+1 supp log m n for all n, there exists an index n 0 such that for all n>n 0 and all n supp log m n, we have n m n+1 and (log m n ) mn+1 =±1. x +e logx+e loglogx+e x +e logx+e loglogx+e +loglogx+loglogx +log x
53 Surreal numbers as transseries 25/28 Definition A transserial derivation on T is a derivation : T T such that TD1. is strong (i.e., it preserves infinite summation); TD2. log f = f / f for all f T >0 ; TD3. nested transseries are differentiated in the natural way.
54 Surreal numbers as transseries 25/28 Definition A transserial derivation on T is a derivation : T T such that TD1. is strong (i.e., it preserves infinite summation); TD2. log f = f / f for all f T >0 ; TD3. nested transseries are differentiated in the natural way. Theorem (BERARDUCCI MANTOVA, 2015) No is a field of transseries and BM is a transserial derivation.
55 Surreal numbers as transseries 25/28 Definition A transserial derivation on T is a derivation : T T such that TD1. is strong (i.e., it preserves infinite summation); TD2. log f = f / f for all f T >0 ; TD3. nested transseries are differentiated in the natural way. Theorem (BERARDUCCI MANTOVA, 2015) No is a field of transseries and BM is a transserial derivation. Corollary Any H-field with constant field R can be embedded in a field of transseries with a transserial derivation.
56 What next? 26/28 (surreal) Numbers H-fields Transseries Germs (in HARDY fields)
57 What next? 26/28 (surreal) Numbers beyond H-fields Transseries Germs (in HARDY fields)
58 What next? 26/28 (surreal) Numbers beyond H-fields Hyperseries Germs (in HARDY fields)
59 Hyperseries = Surreal numbers 27/28 Solving functional equations lack of hyperlogarithms log ω log x = log ω x 1
60 Hyperseries = Surreal numbers 27/28 Solving functional equations lack of hyperlogarithms log ω log x = log ω x 1 log ω x = 1 x log x log log x
61 Hyperseries = Surreal numbers 27/28 Solving functional equations lack of hyperlogarithms log ω log x = log ω x 1 log α x = β<α 1 log β x
62 Hyperseries = Surreal numbers 27/28 Solving functional equations lack of hyperlogarithms Problem with BM log ω log x = log ω x 1 log α x = β<α 1 log β x BM (exp ω (exp ω ω)) = exp ω (exp ω x) exp ω (exp ω x)exp ω x
63 Hyperseries = Surreal numbers 27/28 Solving functional equations lack of hyperlogarithms Problem with BM Conjecture log ω log x = log ω x 1 log α x = β<α 1 log β x BM (exp ω (exp ω ω)) = exp ω (exp ω x) exp ω (exp ω x)exp ω x For a suitable definition of the class Hy of hyperseries (including the nested ones), we have No Hy for the map φ: Hy No; f f (ω).
64 Thank you!
Surreal numbers, derivations and transseries
Vincenzo Mantova Joint work with A. Berarducci University of CamerinoScuola Normale Superiore, Pisa Paris 12 January 2014 Outline 1 Surreal numbers 2 Hardy fields and transseries 3 Surreal derivations
More informationValued Differential Fields. Matthias Aschenbrenner
Valued Differential Fields Matthias Aschenbrenner Overview I ll report on joint work with LOU VAN DEN DRIES and JORIS VAN DER HOEVEN. About three years ago we achieved decisive results on the model theory
More informationarxiv: v1 [math.lo] 18 Nov 2017
arxiv:1711.06936v1 [math.lo] 18 Nov 2017 On Numbers, Germs, and Transseries Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven Abstract Germs of real-valued functions, surreal numbers, and
More informationTowards a Model Theory for Transseries
Towards a Model Theory for Transseries Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven For Anand Pillay, on his 60th birthday. Abstract The differential field of transseries extends
More informationMotivation Automatic asymptotics and transseries Asymptotics of non linear phenomena. Systematic theorie. Eective theory ) emphasis on algebraic aspec
Introduction to automatic asymptotics by Joris van der Hoeven CNRS Universite d'orsay, France email: vdhoeven@math.u-psud.fr web: http://ultralix.polytechnique.fr/~vdhoeven MSRI, Berkeley, 6-0-998 Motivation
More informationRecursive definitions on surreal numbers
Recursive definitions on surreal numbers Antongiulio Fornasiero 19th July 2005 Abstract Let No be Conway s class of surreal numbers. I will make explicit the notion of a function f on No recursively defined
More informationAn introduction to o-minimality
An introduction to o-minimality Notes by Tom Foster and Marcello Mamino on lectures given by Alex Wilkie at the MODNET summer school, Berlin, September 2007 1 Introduction Throughout these notes definable
More informationDIMENSION IN THE REALM OF TRANSSERIES
DIMENSION IN THE REALM OF TRANSSERIES MATTHIAS ASCHENBRENNER, LOU VAN DEN DRIES, AND JORIS VAN DER HOEVEN Abstract. Let T be the differential field of transseries. We establish some basic properties of
More informationHardy type derivations on generalised series fields
Hardy type derivations on generalised series fields Salma Kuhlmann. Universität Konstanz, Fachbereich Mathematik und Statistik 78457 Konstanz, Germany. Email: salma.kuhlmann@uni-konstanz.de Webpage: http://www.math.uni-konstanz.de/
More informationTRUNCATION IN HAHN FIELDS
TRUNCATION IN HAHN FIELDS LOU VAN DEN DRIES 1. Introduction Let k be a field and Γ an (additive) ordered 1 abelian group. Then we have the Hahn field K := k((t Γ )) whose elements are the formal series
More informationClosed Asymptotic Couples
Closed Asymptotic Couples Matthias Aschenbrenner and Lou van den Dries * University of Illinois at Urbana-Champaign, Department of Mathematics Urbana, IL 61801, U.S.A. E-mail: maschenb@math.uiuc.edu; vddries@math.uiuc.edu
More informationECEN 5682 Theory and Practice of Error Control Codes
ECEN 5682 Theory and Practice of Error Control Codes Introduction to Algebra University of Colorado Spring 2007 Motivation and For convolutional codes it was convenient to express the datawords and the
More informationDefinable Extension Theorems in O-minimal Structures. Matthias Aschenbrenner University of California, Los Angeles
Definable Extension Theorems in O-minimal Structures Matthias Aschenbrenner University of California, Los Angeles 1 O-minimality Basic definitions and examples Geometry of definable sets Why o-minimal
More informationPart II. Logic and Set Theory. Year
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 60 Paper 4, Section II 16G State and prove the ǫ-recursion Theorem. [You may assume the Principle of ǫ- Induction.]
More informationPart II Logic and Set Theory
Part II Logic and Set Theory Theorems Based on lectures by I. B. Leader Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationDifferentially algebraic gaps
Sel. math., New ser. Online First c 2005 Birkhäuser Verlag Basel/Switzerland DOI 10.1007/s00029-005-0010-0 Selecta Mathematica New Series Differentially algebraic gaps Matthias Aschenbrenner, Lou van den
More informationA differential intermediate value theorem
A differential intermediate value theorem by Joris van der Hoeven Dépt. de Mathématiques (Bât. 425) Université Paris-Sud 91405 Orsay Cedex France June 2000 Abstract Let T be the field of grid-based transseries
More informationFields and model-theoretic classification, 2
Fields and model-theoretic classification, 2 Artem Chernikov UCLA Model Theory conference Stellenbosch, South Africa, Jan 11 2017 NIP Definition Let T be a complete first-order theory in a language L.
More informationω-stable Theories: Introduction
ω-stable Theories: Introduction 1 ω - Stable/Totally Transcendental Theories Throughout let T be a complete theory in a countable language L having infinite models. For an L-structure M and A M let Sn
More informationChapter 8. P-adic numbers. 8.1 Absolute values
Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.
More informationLöwenheim-Skolem Theorems, Countable Approximations, and L ω. David W. Kueker (Lecture Notes, Fall 2007)
Löwenheim-Skolem Theorems, Countable Approximations, and L ω 0. Introduction David W. Kueker (Lecture Notes, Fall 2007) In its simplest form the Löwenheim-Skolem Theorem for L ω1 ω states that if σ L ω1
More informationOn expansions of the real field by complex subgroups
On expansions of the real field by complex subgroups University of Illinois at Urbana-Champaign July 21, 2016 Previous work Expansions of the real field R by subgroups of C have been studied previously.
More information1 of 8 7/15/2009 3:43 PM Virtual Laboratories > 1. Foundations > 1 2 3 4 5 6 7 8 9 6. Cardinality Definitions and Preliminary Examples Suppose that S is a non-empty collection of sets. We define a relation
More informationNIP FOR THE ASYMPTOTIC COUPLE OF THE FIELD OF LOGARITHMIC TRANSSERIES
NIP FOR THE ASYMPTOTIC COUPLE OF THE FIELD OF LOGARITHMIC TRANSSERIES ALLEN GEHRET Abstract. The derivation on the differential-valued field T log of logarithmic transseries induces on its value group
More informationClassifying classes of structures in model theory
Classifying classes of structures in model theory Saharon Shelah The Hebrew University of Jerusalem, Israel, and Rutgers University, NJ, USA ECM 2012 Saharon Shelah (HUJI and Rutgers) Classifying classes
More informationIntroduction to Kleene Algebras
Introduction to Kleene Algebras Riccardo Pucella Basic Notions Seminar December 1, 2005 Introduction to Kleene Algebras p.1 Idempotent Semirings An idempotent semiring is a structure S = (S, +,, 1, 0)
More informationCounterexamples to witness conjectures. Notations Let E be the set of admissible constant expressions built up from Z; + ;?;;/; exp and log. Here a co
Counterexamples to witness conjectures Joris van der Hoeven D pt. de Math matiques (b t. 45) Universit Paris-Sud 91405 Orsay CEDEX France August 15, 003 Consider the class of exp-log constants, which is
More informationOn Computably Enumerable Sets over Function Fields
On Computably Enumerable Sets over Function Fields Alexandra Shlapentokh East Carolina University Joint Meetings 2017 Atlanta January 2017 Some History Outline 1 Some History A Question and the Answer
More information15. Polynomial rings Definition-Lemma Let R be a ring and let x be an indeterminate.
15. Polynomial rings Definition-Lemma 15.1. Let R be a ring and let x be an indeterminate. The polynomial ring R[x] is defined to be the set of all formal sums a n x n + a n 1 x n +... a 1 x + a 0 = a
More informationtp(c/a) tp(c/ab) T h(m M ) is assumed in the background.
Model Theory II. 80824 22.10.2006-22.01-2007 (not: 17.12) Time: The first meeting will be on SUNDAY, OCT. 22, 10-12, room 209. We will try to make this time change permanent. Please write ehud@math.huji.ac.il
More informationFINITE MODEL THEORY (MATH 285D, UCLA, WINTER 2017) LECTURE NOTES IN PROGRESS
FINITE MODEL THEORY (MATH 285D, UCLA, WINTER 2017) LECTURE NOTES IN PROGRESS ARTEM CHERNIKOV 1. Intro Motivated by connections with computational complexity (mostly a part of computer scientice today).
More informationProperties of O-minimal Exponential Fields
1 Universität Konstanz, Fachbereich Mathematik und Statistik 11 September 2017 19th ÖMG Congress and Annual DMV Meeting 1 Supported by DVMLG-Nachwuchsstipendium 1 Preliminary Notions 2 Decidability of
More informationModel Theory MARIA MANZANO. University of Salamanca, Spain. Translated by RUY J. G. B. DE QUEIROZ
Model Theory MARIA MANZANO University of Salamanca, Spain Translated by RUY J. G. B. DE QUEIROZ CLARENDON PRESS OXFORD 1999 Contents Glossary of symbols and abbreviations General introduction 1 xix 1 1.0
More informationExploring the infinite : an introduction to proof and analysis / Jennifer Brooks. Boca Raton [etc.], cop Spis treści
Exploring the infinite : an introduction to proof and analysis / Jennifer Brooks. Boca Raton [etc.], cop. 2017 Spis treści Preface xiii I Fundamentals of Abstract Mathematics 1 1 Basic Notions 3 1.1 A
More informationThe triple helix. John R. Steel University of California, Berkeley. October 2010
The triple helix John R. Steel University of California, Berkeley October 2010 Three staircases Plan: I. The interpretability hierarchy. II. The vision of ultimate K. III. The triple helix. IV. Some locator
More informationProof Theory and Subsystems of Second-Order Arithmetic
Proof Theory and Subsystems of Second-Order Arithmetic 1. Background and Motivation Why use proof theory to study theories of arithmetic? 2. Conservation Results Showing that if a theory T 1 proves ϕ,
More informationA MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ
A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In
More informationTame definable topological dynamics
Tame definable topological dynamics Artem Chernikov (Paris 7) Géométrie et Théorie des Modèles, 4 Oct 2013, ENS, Paris Joint work with Pierre Simon, continues previous work with Anand Pillay and Pierre
More information2 Section 2 However, in order to apply the above idea, we will need to allow non standard intervals ('; ) in the proof. More precisely, ' and may gene
Introduction 1 A dierential intermediate value theorem by Joris van der Hoeven D pt. de Math matiques (B t. 425) Universit Paris-Sud 91405 Orsay Cedex France June 2000 Abstract Let T be the eld of grid-based
More informationECEN 5022 Cryptography
Elementary Algebra and Number Theory University of Colorado Spring 2008 Divisibility, Primes Definition. N denotes the set {1, 2, 3,...} of natural numbers and Z denotes the set of integers {..., 2, 1,
More informationPRESERVATION THEOREMS IN LUKASIEWICZ MODEL THEORY
Iranian Journal of Fuzzy Systems Vol. 10, No. 3, (2013) pp. 103-113 103 PRESERVATION THEOREMS IN LUKASIEWICZ MODEL THEORY S. M. BAGHERI AND M. MONIRI Abstract. We present some model theoretic results for
More informationLecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman
Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman October 17, 2006 TALK SLOWLY AND WRITE NEATLY!! 1 0.1 Integral Domains and Fraction Fields 0.1.1 Theorems Now what we are going
More informationSolutions to Unique Readability Homework Set 30 August 2011
s to Unique Readability Homework Set 30 August 2011 In the problems below L is a signature and X is a set of variables. Problem 0. Define a function λ from the set of finite nonempty sequences of elements
More informationFields and Galois Theory. Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory.
Fields and Galois Theory Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory. This should be a reasonably logical ordering, so that a result here should
More informationIntroduction to Model Theory
Introduction to Model Theory Charles Steinhorn, Vassar College Katrin Tent, University of Münster CIRM, January 8, 2018 The three lectures Introduction to basic model theory Focus on Definability More
More informationComputable Transformations of Structures
Computable Transformations of Structures Russell Miller Queens College C.U.N.Y., 65-30 Kissena Blvd. Queens NY 11367 USA Graduate Center of C.U.N.Y., 365 Fifth Avenue New York, NY 10016 USA qcpages.qc.cuny.edu/
More informationAn effective version of Wilkie s theorem of the complement and some effective o-minimality results
An effective version of Wilkie s theorem of the complement and some effective o-minimality results Alessandro Berarducci Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa Tamara
More informationExistence of a Limit on a Dense Set, and. Construction of Continuous Functions on Special Sets
Existence of a Limit on a Dense Set, and Construction of Continuous Functions on Special Sets REU 2012 Recap: Definitions Definition Given a real-valued function f, the limit of f exists at a point c R
More informationPart VI. Extension Fields
VI.29 Introduction to Extension Fields 1 Part VI. Extension Fields Section VI.29. Introduction to Extension Fields Note. In this section, we attain our basic goal and show that for any polynomial over
More informationVAUGHT S THEOREM: THE FINITE SPECTRUM OF COMPLETE THEORIES IN ℵ 0. Contents
VAUGHT S THEOREM: THE FINITE SPECTRUM OF COMPLETE THEORIES IN ℵ 0 BENJAMIN LEDEAUX Abstract. This expository paper introduces model theory with a focus on countable models of complete theories. Vaught
More informationOctober 12, Complexity and Absoluteness in L ω1,ω. John T. Baldwin. Measuring complexity. Complexity of. concepts. to first order.
October 12, 2010 Sacks Dicta... the central notions of model theory are absolute absoluteness, unlike cardinality, is a logical concept. That is why model theory does not founder on that rock of undecidability,
More informationMore Model Theory Notes
More Model Theory Notes Miscellaneous information, loosely organized. 1. Kinds of Models A countable homogeneous model M is one such that, for any partial elementary map f : A M with A M finite, and any
More informationREVIEW OF ESSENTIAL MATH 346 TOPICS
REVIEW OF ESSENTIAL MATH 346 TOPICS 1. AXIOMATIC STRUCTURE OF R Doğan Çömez The real number system is a complete ordered field, i.e., it is a set R which is endowed with addition and multiplication operations
More informationMODEL THEORY FOR ALGEBRAIC GEOMETRY
MODEL THEORY FOR ALGEBRAIC GEOMETRY VICTOR ZHANG Abstract. We demonstrate how several problems of algebraic geometry, i.e. Ax-Grothendieck, Hilbert s Nullstellensatz, Noether- Ostrowski, and Hilbert s
More informationTRANSCENDENTAL NUMBERS AND PERIODS. Contents
TRANSCENDENTAL NUMBERS AND PERIODS JAMES CARLSON Contents. Introduction.. Diophantine approximation I: upper bounds 2.2. Diophantine approximation II: lower bounds 4.3. Proof of the lower bound 5 2. Periods
More informationAlgebraic Cryptography Exam 2 Review
Algebraic Cryptography Exam 2 Review You should be able to do the problems assigned as homework, as well as problems from Chapter 3 2 and 3. You should also be able to complete the following exercises:
More informationSome Background Material
Chapter 1 Some Background Material In the first chapter, we present a quick review of elementary - but important - material as a way of dipping our toes in the water. This chapter also introduces important
More informationUniversal Totally Ordered Sets
Universal Totally Ordered Sets Luke Adams May 11, 2018 Abstract In 1874, Georg Cantor published an article in which he proved that the set of algebraic numbers are countable, and the set of real numbers
More informationarxiv:math.lo/ v1 28 Nov 2004
HILBERT SPACES WITH GENERIC GROUPS OF AUTOMORPHISMS arxiv:math.lo/0411625 v1 28 Nov 2004 ALEXANDER BERENSTEIN Abstract. Let G be a countable group. We proof that there is a model companion for the approximate
More informationLimit computable integer parts
Limit computable integer parts Paola D Aquino, Julia Knight, and Karen Lange July 12, 2010 Abstract Let R be a real closed field. An integer part I for R is a discretely ordered subring such that for every
More informationINTRODUCTION TO CARDINAL NUMBERS
INTRODUCTION TO CARDINAL NUMBERS TOM CUCHTA 1. Introduction This paper was written as a final project for the 2013 Summer Session of Mathematical Logic 1 at Missouri S&T. We intend to present a short discussion
More informationSpring 2014 Advanced Probability Overview. Lecture Notes Set 1: Course Overview, σ-fields, and Measures
36-752 Spring 2014 Advanced Probability Overview Lecture Notes Set 1: Course Overview, σ-fields, and Measures Instructor: Jing Lei Associated reading: Sec 1.1-1.4 of Ash and Doléans-Dade; Sec 1.1 and A.1
More informationDO FIVE OUT OF SIX ON EACH SET PROBLEM SET
DO FIVE OUT OF SIX ON EACH SET PROBLEM SET 1. THE AXIOM OF FOUNDATION Early on in the book (page 6) it is indicated that throughout the formal development set is going to mean pure set, or set whose elements,
More informationThe Axiom of Infinity, Quantum Field Theory, and Large Cardinals. Paul Corazza Maharishi University
The Axiom of Infinity, Quantum Field Theory, and Large Cardinals Paul Corazza Maharishi University The Quest for an Axiomatic Foundation For Large Cardinals Gödel believed natural axioms would be found
More informationSUPPLEMENT TO DENSE PAIRS OF O-MINIMAL STRUCTURES BY LOU VAN DEN DRIES
SUPPLEMENT TO DENSE PIRS OF O-MINIML STRUCTURES BY LOU VN DEN DRIES LLEN GEHRET Introduction If in doubt, all notations and assumptions are with respect to [7]. Generally speaking, exercises are things
More informationA finite universal SAGBI basis for the kernel of a derivation. Osaka Journal of Mathematics. 41(4) P.759-P.792
Title Author(s) A finite universal SAGBI basis for the kernel of a derivation Kuroda, Shigeru Citation Osaka Journal of Mathematics. 4(4) P.759-P.792 Issue Date 2004-2 Text Version publisher URL https://doi.org/0.890/838
More informationConstruction of a general measure structure
Chapter 4 Construction of a general measure structure We turn to the development of general measure theory. The ingredients are a set describing the universe of points, a class of measurable subsets along
More informationDefinably amenable groups in NIP
Definably amenable groups in NIP Artem Chernikov (Paris 7) Lyon, 21 Nov 2013 Joint work with Pierre Simon. Setting T is a complete first-order theory in a language L, countable for simplicity. M = T a
More informationShort notes on Axioms of set theory, Well orderings and Ordinal Numbers
Short notes on Axioms of set theory, Well orderings and Ordinal Numbers August 29, 2013 1 Logic and Notation Any formula in Mathematics can be stated using the symbols and the variables,,,, =, (, ) v j
More informationHandbook of Logic and Proof Techniques for Computer Science
Steven G. Krantz Handbook of Logic and Proof Techniques for Computer Science With 16 Figures BIRKHAUSER SPRINGER BOSTON * NEW YORK Preface xvii 1 Notation and First-Order Logic 1 1.1 The Use of Connectives
More informationZILBER S PSEUDO-EXPONENTIAL FIELDS. The goal of this talk is to prove the following facts above Zilber s pseudoexponential
ZILBER S PSEUDO-EXPONENTIAL FIELDS WILL BONEY The goal of this talk is to prove the following facts above Zilber s pseudoexponential fields (1) they are axiomatized in L ω1,ω(q) and this logic is essential.
More informationNOTES ON WELL ORDERING AND ORDINAL NUMBERS. 1. Logic and Notation Any formula in Mathematics can be stated using the symbols
NOTES ON WELL ORDERING AND ORDINAL NUMBERS TH. SCHLUMPRECHT 1. Logic and Notation Any formula in Mathematics can be stated using the symbols,,,, =, (, ) and the variables v j : where j is a natural number.
More information4 Countability axioms
4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said
More informationvan Rooij, Schikhof: A Second Course on Real Functions
vanrooijschikhof.tex April 25, 2018 van Rooij, Schikhof: A Second Course on Real Functions Notes from [vrs]. Introduction A monotone function is Riemann integrable. A continuous function is Riemann integrable.
More informationApplications of model theory in extremal graph combinatorics
Applications of model theory in extremal graph combinatorics Artem Chernikov (IMJ-PRG, UCLA) Logic Colloquium Helsinki, August 4, 2015 Szemerédi regularity lemma Theorem [E. Szemerédi, 1975] Every large
More informationUnsolvable problems, the Continuum Hypothesis, and the nature of infinity
Unsolvable problems, the Continuum Hypothesis, and the nature of infinity W. Hugh Woodin Harvard University January 9, 2017 V : The Universe of Sets The power set Suppose X is a set. The powerset of X
More informationAn inner model from Ω-logic. Daisuke Ikegami
An inner model from Ω-logic Daisuke Ikegami Kobe University 12. November 2014 Goal & Result Goal Construct a model of set theory which is close to HOD, but easier to analyze. Goal & Result Goal Construct
More informationFormal power series rings, inverse limits, and I-adic completions of rings
Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More informationMATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. II - Model Theory - H. Jerome Keisler
ATHEATCS: CONCEPTS, AND FOUNDATONS Vol. - odel Theory - H. Jerome Keisler ODEL THEORY H. Jerome Keisler Department of athematics, University of Wisconsin, adison Wisconsin U.S.A. Keywords: adapted probability
More informationDecidable and undecidable real closed rings
Decidable and undecidable real closed rings Marcus Tressl marcus.tressl@manchester.ac.uk http://personalpages.manchester.ac.uk/staff/marcus.tressl/index.php The University of Manchester Antalya Algebra
More informationPure Patterns and Ordinal Numbers
Pure Patterns and Ordinal Numbers Gunnar Wilken Okinawa Institute of Science and Technology Graduate University Japan Sets and Computations Institute of the Mathematical Sciences National University of
More informationDIOPHANTINE APPROXIMATION AND CONTINUED FRACTIONS IN POWER SERIES FIELDS
DIOPHANTINE APPROXIMATION AND CONTINUED FRACTIONS IN POWER SERIES FIELDS A. LASJAUNIAS Avec admiration pour Klaus Roth 1. Introduction and Notation 1.1. The Fields of Power Series F(q) or F(K). Let p be
More information(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d
The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers
More informationPartial Collapses of the Σ 1 Complexity Hierarchy in Models for Fragments of Bounded Arithmetic
Partial Collapses of the Σ 1 Complexity Hierarchy in Models for Fragments of Bounded Arithmetic Zofia Adamowicz Institute of Mathematics, Polish Academy of Sciences Śniadeckich 8, 00-950 Warszawa, Poland
More informationWinter School on Galois Theory Luxembourg, February INTRODUCTION TO PROFINITE GROUPS Luis Ribes Carleton University, Ottawa, Canada
Winter School on alois Theory Luxembourg, 15-24 February 2012 INTRODUCTION TO PROFINITE ROUPS Luis Ribes Carleton University, Ottawa, Canada LECTURE 2 2.1 ENERATORS OF A PROFINITE ROUP 2.2 FREE PRO-C ROUPS
More informationModel Theory and o-minimal Structures
Model Theory and o-minimal Structures Jung-Uk Lee Dept. Math. Yonsei University August 3, 2010 1 / 41 Outline 1 Inroduction to Model Theory What is model theory? Language, formula, and structures Theory
More informationMathematisches Forschungsinstitut Oberwolfach. Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations
Mathematisches Forschungsinstitut Oberwolfach Report No. 60/2016 DOI: 10.4171/OWR/2016/60 Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations Organised by Alessandro Berarducci,
More informationJónsson posets and unary Jónsson algebras
Jónsson posets and unary Jónsson algebras Keith A. Kearnes and Greg Oman Abstract. We show that if P is an infinite poset whose proper order ideals have cardinality strictly less than P, and κ is a cardinal
More information2.2 Lowenheim-Skolem-Tarski theorems
Logic SEP: Day 1 July 15, 2013 1 Some references Syllabus: http://www.math.wisc.edu/graduate/guide-qe Previous years qualifying exams: http://www.math.wisc.edu/ miller/old/qual/index.html Miller s Moore
More informationHomework in Topology, Spring 2009.
Homework in Topology, Spring 2009. Björn Gustafsson April 29, 2009 1 Generalities To pass the course you should hand in correct and well-written solutions of approximately 10-15 of the problems. For higher
More informationExtension theorems for homomorphisms
Algebraic Geometry Fall 2009 Extension theorems for homomorphisms In this note, we prove some extension theorems for homomorphisms from rings to algebraically closed fields. The prototype is the following
More informationTOPOLOGICAL DIFFERENTIAL FIELDS
TOPOLOGICAL DIFFERENTIAL FIELDS NICOLAS GUZY AND FRANÇOISE POINT Abstract. First, we consider first-order theories of topological fields admitting a model-completion and their expansion to differential
More informationPlaces of Number Fields and Function Fields MATH 681, Spring 2018
Places of Number Fields and Function Fields MATH 681, Spring 2018 From now on we will denote the field Z/pZ for a prime p more compactly by F p. More generally, for q a power of a prime p, F q will denote
More informationTHE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS. K. R. Goodearl and E. S. Letzter
THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS K. R. Goodearl and E. S. Letzter Abstract. In previous work, the second author introduced a topology, for spaces of irreducible representations,
More informationADVANCE TOPICS IN ANALYSIS - REAL. 8 September September 2011
ADVANCE TOPICS IN ANALYSIS - REAL NOTES COMPILED BY KATO LA Introductions 8 September 011 15 September 011 Nested Interval Theorem: If A 1 ra 1, b 1 s, A ra, b s,, A n ra n, b n s, and A 1 Ě A Ě Ě A n
More informationWhat is the right type-space? Humboldt University. July 5, John T. Baldwin. Which Stone Space? July 5, Tameness.
Goals The fundamental notion of a Stone space is delicate for infinitary logic. I will describe several possibilities. There will be a quiz. Infinitary Logic and Omitting Types Key Insight (Chang, Lopez-Escobar)
More informationFactorization in Polynomial Rings
Factorization in Polynomial Rings Throughout these notes, F denotes a field. 1 Long division with remainder We begin with some basic definitions. Definition 1.1. Let f, g F [x]. We say that f divides g,
More informationContents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3
Analysis Math Notes Study Guide Real Analysis Contents Ordered Fields 2 Ordered sets and fields 2 Construction of the Reals 1: Dedekind Cuts 2 Metric Spaces 3 Metric Spaces 3 Definitions 4 Separability
More informationWEIERSTRASS THEOREMS AND RINGS OF HOLOMORPHIC FUNCTIONS
WEIERSTRASS THEOREMS AND RINGS OF HOLOMORPHIC FUNCTIONS YIFEI ZHAO Contents. The Weierstrass factorization theorem 2. The Weierstrass preparation theorem 6 3. The Weierstrass division theorem 8 References
More information