On Numbers, Germs, and Transseries

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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

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