On Numbers, Germs, and Transseries

Matthias Aschenbrenner Lou van den Dries Joris van der Hoeven UCLA UIUC CNRS Introduction 2/28

Three intimately related topics... (surreal) Numbers

Transseries

Germs (in HARDY fields) Introduction 2/28

Germs (in HARDY fields) HARDY fields 3/28

Let C 1 be the of germs at +∞ of continuously differentiable functions (a,∞)→ℝ (a∈ℝ). We denote the germ at +∞ of a 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 2 ℚ, ℝ, ℝ(x), ℝ(x,ex), ℝ(x,ex,log x), ℝx,ex ,erf x 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 1 f (t)>0, eventually, or f ≠0 ⟹ ∈H ⟹ { f { f (t)<0, eventually. Consequently, • H carries an ordering making H an ordered field: f >0 ⟺ f (t)>0 eventually; • f is eventually monotonic, and lim f (t)∈ℝ ∪{±∞}. t→+∞ Transseries 5/28

(surreal) Numbers

Transseries

Germs (in HARDY fields) Transseries 5/28

Transseries The field 핋 of transseries 6/28

핋≔ closure of ℝ ∪{x} under exp, log and infinite summation

x 2 ee +⋯ − 3ex +5(log x)π +42+x−1 +⋯+e−x Here one should think of x as a positive infinite indeterminate. The field 핋 of transseries 6/28

핋≔ closure of ℝ ∪{x} under exp, log and infinite summation

x x/2 2 ee +e +⋯ − 3ex +5(log x)π +42+x−1 +2x−2 +⋯+e−x Here one should think of x as a positive infinite indeterminate. The field 핋 of transseries 6/28

핋≔ closure of ℝ ∪{x} under exp, log and infinite summation

x x/2 x/3 2 ee +e +e +⋯ − 3ex +5(log x)π +42+x−1 +2x−2 +6x−3 +⋯+e−x Here one should think of x as a positive infinite indeterminate. The field 핋 of transseries 6/28

핋≔ closure of ℝ ∪{x} under exp, log and infinite summation

x x/2 x/3 2 ee +e +e +⋯ − 3ex +5(log x)π +42+x−1 +2x−2 +6x−3 +24x−4 +⋯+e−x Here one should think of x as a positive infinite indeterminate. The field 핋 of transseries 6/28

핋≔ closure of ℝ ∪{x} under exp, log and infinite summation

ex+ex/2+⋯ x2 π −1 −2 −3 −x f픪 픪 = e − 3e +5(log x) +42+x +2x +6x +⋯+e 픪 x: positive infinite indeterminate f픪: coefficent 픪: transmonomial The field 핋 of transseries 6/28

핋≔ closure of ℝ ∪{x} under exp, log and infinite summation

ex+ex/2+⋯ x2 π −1 −2 −3 −x f픪 픪 = e − 3e +5(log x) +42+x +2x +6x +⋯+e 픪 x: positive infinite indeterminate f픪: coefficent 픪: transmonomial

The formal definition of 핋 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 핋: 1 1 1 1 1 1 1 + x + x + ex +⋯ + + +⋯ x e ee ee x xlog x xlog xlog log x 핋 as an ordered differential field 7/28

• With the natural ordering of transseries (via the leading coefficient), 핋 is a real closed ordered field extension of ℝ. • Each f ∈핋 can be differentiated term by term (with x′=1):

∞ ex ′ ∞ ex ′ ∞ ex ex ex ( n! ) = n! = n! − n = ( xn ) (xn ) (xn xn+1 ) x (n=0 ) n=0 ( ) n=0 ( ) • This yields a derivation f ↦ f ′ on the field 핋:

( f + g)′= f ′+ g′, ( f ⋅ g)′= f ′⋅ g+ f ⋅ g′

Its constant field is { f ∈핋: f ′=0} =ℝ. • Given f , g∈핋, the equation y′+ fy = g admits a solution y ≠0 in 핋. Surreal numbers 8/28

(surreal) Numbers

Transseries

Germs (in HARDY fields) Surreal numbers 8/28

(surreal) Numbers 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. Addition of surreal numbers 10/28

• x∈No is simpler than y ∈No :⟺ x is a prefix of y • For L

x+y ≔ {xL +y,x+yL |xR +y,x+yR}.

(Idea: we want xL +y < x+y < xR +y, …) Exponentiation and differentiation 11/28

• In the 1980s, GONSHOR (based on ideas of KRUSKAL) defined an expo- nential function exp:No→No>0 that extends x↦ex on ℝ.

• In 2006, BERARDUCCI and MANTOVA (using ideas of VDH and SCHMELING) defined a derivation ∂BM on No with

ker ∂BM =ℝ, ∂BM(휔)=1, ∂BM (exp( f ))=∂BM( f )⋅exp( f ) for f ∈No. In a certain technical sense, it is the simplest such derivation that satis- fies some natural further conditions. • The BM-derivation on No behaves in many ways like the derivation 1 on 핋, with 휔 >ℝ in the role of x>ℝ. For instance, ∂BM(log 휔)= 휔 . Towards a unified theory 12/28

(surreal) Numbers

Transseries

Germs (in HARDY fields) Towards a unified theory 12/28

(surreal) Numbers

? Transseries Germs (in HARDY fields) Towards a unified theory 12/28

(surreal) Numbers

H-fields Transseries

Germs (in HARDY fields) Towards a unified theory 12/28

Hardy (surreal) Hausdorff Numbers Hahn

H-fields Transseries

Germs (in HARDY fields) 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 x In 핋: 0 ≺ e−x ≺ x−10 ≺ 1 ≍ 100 ≺ log x ≺ x1/10 ≺ ex ∼ex +x ≺ ee H-fields 14/28

Definition We call K an H-field if H1. f >C ⟹ f ′>0; H2. f ≍1 ⟹ f ∼c for some c∈C.

Examples HARDY fields containing ℝ; ordered differential subfields of 핋 or No that contain ℝ. 핋 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. One of our main results 15/28

We view 핋 model-theoretically as a structure with the primitives 0, 1, +, ×, ∂ (derivation), ⩽ (ordering).

Theorem (Ann. of Math. Stud. vol. 195) The elementary theory of 핋 is completely axiomatized by: •1 핋 is a LIOUVILLE closed H-field with small derivation; •2 핋 satisfies the intermediate value property for differential polynomials. Actually •2 is a bit of an afterthought. A corollary of the theorem: the theory of 핋 is decidable. We also prove a quantifier elimination result for 핋 in a natural expansion of the above language. H-field elements as germs 16/28

(surreal) Numbers

H-fields Transseries

Germs (in HARDY fields) H-field elements as germs 16/28

(surreal) Numbers

Transseries

Germs H-fields (in HARDY fields) Closure properties of HARDY fields 17/28

Theorem (HARDY 1910, BOURBAKI 1951) Any HARDY field has a smallest LIOUVILLE closed HARDY field extension. 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 . If y∈ C 1 satisfies the differential equation y′=Φ(y), then H⟨y⟩ =H(y,y′) is a HARDY field. 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 2 y′′+y =ex lies in a HARDY field, but any HARDY field contains at most one solution. Conjectural closure properties 18/28

Conjecture Let H be a maximal HARDY field. Then •A H satisfies the differential intermediate value property. •B For countable subsets A

Conjecture Let H be a maximal HARDY field. Then •A H satisfies the differential intermediate value property. •B For countable subsets A

Corollary •A H is elementarily equivalent to 핋 as an ordered differential field. •B Under CH, all maximal HARDY fields are isomorphic. Work in progress on Conjecture A 19/28

Theorem (VAN DER HOEVEN 2009) The subfield 핋da of transseries that satisfy an algebraic differential equation over ℝ can be embedded (as an ordered differential field) in a HARDY field. Work in progress on Conjecture A 19/28

Theorem (VAN DER HOEVEN 2009) The subfield 핋da of transseries that satisfy an algebraic differential equation over ℝ can be embedded (as an ordered differential field) in a HARDY field.

x y′′ = e−e +y2 Work in progress on Conjecture A 19/28

Theorem (VAN DER HOEVEN 2009) The subfield 핋da of transseries that satisfy an algebraic differential equation over ℝ can be embedded (as an ordered differential field) in a HARDY field.

x y = (e−e +y2)

x x 2 x 3 = e−e + e−e +2 e−e +⋯ Work in progress on Conjecture A 19/28

Theorem (VAN DER HOEVEN 2009) The subfield 핋da of transseries that satisfy an algebraic differential equation over ℝ can be embedded (as an ordered differential field) in a HARDY field.

x y = (e−e +y2)

x x 2 x 3 = e−e + e−e +2 e−e +⋯

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. Work in progress on Conjecture B 20/28

Theorem (BOREL 1895) Any y ∈ℝ[[x−1]] is the asymptotic expansion of a germ y˜ in C ∞. Work in progress on Conjecture B 20/28

Theorem (BOREL 1895) Any y ∈ℝ[[x−1]] is the asymptotic expansion of a germ y˜ in C ∞.

Example y =x−1 +2!!x−2 +3!!x−3 +⋯ is differentially transcendental over ℝ ⟹ the differential field ℝ⟨y˜⟩ =ℝ(y˜,y˜′,y˜′′,…) is a HARDY field. Work in progress on Conjecture B 20/28

Theorem (BOREL 1895) Any y ∈ℝ[[x−1]] is the asymptotic expansion of a germ y˜ in C ∞.

Example y =x−1 +2!!x−2 +3!!x−3 +⋯ is differentially transcendental over ℝ ⟹ the differential field ℝ⟨y˜⟩ =ℝ(y˜,y˜′,y˜′′,…) is a HARDY field.

Theorem in progress

✓ Every pseudocauchy sequence (yn) 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. H-field elements as surreal numbers 21/28

(surreal) Numbers

H-fields Transseries

Germs (in HARDY fields) H-field elements as surreal numbers 21/28

(surreal) Numbers H-fields

Transseries

Germs (in HARDY fields) 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. 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 휅 be an uncountable cardinal. The field No(휅) of surreal numbers of length <휅 is an elementary submodel of No. 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 휅 be an uncountable cardinal. The field No(휅) of surreal numbers of length <휅 is an elementary submodel of No.

Corollary in progress

Under CH all maximal HARDY fields are isomorphic to No(휔1). H-field elements as transseries 23/28

(surreal) Numbers

H-fields Transseries

Germs (in HARDY fields) H-field elements as transseries 23/28

(surreal) Numbers

H-fields Transseries Germs (in HARDY fields) H-field elements as transseries 23/28

H-fields (surreal) Transseries Numbers Germs (in HARDY fields) How to generalize transseries 24/28

Definition (VAN DER HOEVEN 2000, SCHMELING 2001) A field 퐓 =ℝ[[픐]] with partial log:퐓 −−→− 퐓 is a field of transseries if T1. The domain of log is 퐓>0; T2. for each 픪∈픐 and 픫∈supp log 픪, we have 픫≻1; 1 2 1 3 T3. log (1+휀)=휀− 2 휀 + 3 휀 +⋯, for all 휀∈퐓 with 휀≺1; ℕ T4. for every sequence (픪n)∈픐 with 픪n+1∈supp log 픪n for all n, there exists an index n0 such that for all n>n0 and all 픫∈supp log 픪n, we have 픫≽픪n+1

and (log 픪n)픪n+1 =±1. How to generalize transseries 24/28

Definition (VAN DER HOEVEN 2000, SCHMELING 2001) A field 퐓 =ℝ[[픐]] with partial log:퐓 −−→− 퐓 is a field of transseries if T1. The domain of log is 퐓>0; T2. for each 픪∈픐 and 픫∈supp log 픪, we have 픫≻1; 1 2 1 3 T3. log (1+휀)=휀− 2 휀 + 3 휀 +⋯, for all 휀∈퐓 with 휀≺1; ℕ T4. for every sequence (픪n)∈픐 with 픪n+1∈supp log 픪n for all n, there exists an index n0 such that for all n>n0 and all 픫∈supp log 픪n, we have 픫≽픪n+1

and (log 픪n)픪n+1 =±1.

loglogx+e⋰ loglogx+e⋰+loglogx √x +elogx+e √x +elogx+e +loglogx +log x How to generalize transseries 24/28

Definition (VAN DER HOEVEN 2000, SCHMELING 2001) A field 퐓 =ℝ[[픐]] with partial log:퐓 −−→− 퐓 is a field of transseries if T1. The domain of log is 퐓>0; T2. for each 픪∈픐 and 픫∈supp log 픪, we have 픫≻1; 1 2 1 3 T3. log (1+휀)=휀− 2 휀 + 3 휀 +⋯, for all 휀∈퐓 with 휀≺1; ℕ T4. for every sequence (픪n)∈픐 with 픪n+1∈supp log 픪n for all n, there exists an index n0 such that for all n>n0 and all 픫∈supp log 픪n, we have 픫≽픪n+1

and (log 픪n)픪n+1 =±1.

loglogx+e⋰ loglogx+e⋰+loglogx √x +elogx+e √x +elogx+e +loglogx +log x How to generalize transseries 24/28

Definition (VAN DER HOEVEN 2000, SCHMELING 2001) A field 퐓 =ℝ[[픐]] with partial log:퐓 −−→− 퐓 is a field of transseries if T1. The domain of log is 퐓>0; T2. for each 픪∈픐 and 픫∈supp log 픪, we have 픫≻1; 1 2 1 3 T3. log (1+휀)=휀− 2 휀 + 3 휀 +⋯, for all 휀∈퐓 with 휀≺1; ℕ T4. for every sequence (픪n)∈픐 with 픪n+1∈supp log 픪n for all n, there exists an index n0 such that for all n>n0 and all 픫∈supp log 픪n, we have 픫≽픪n+1

and (log 픪n)픪n+1 =±1.

⋰ loglogx+e⋰ loglogx+e +loglogx √x +elogx+e √x +elogx+e +loglogx +log x Surreal numbers as transseries 25/28

Definition A transserial derivation on 퐓 is a derivation ∂:퐓 →퐓 such that TD1. ∂ is strong (i.e., it preserves infinite summation); TD2. ∂ log f =∂ f / f for all f ∈퐓>0; TD3. nested transseries are differentiated in the “natural” way. Surreal numbers as transseries 25/28

Definition A transserial derivation on 퐓 is a derivation ∂:퐓 →퐓 such that TD1. ∂ is strong (i.e., it preserves infinite summation); TD2. ∂ log f =∂ f / f for all f ∈퐓>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. Surreal numbers as transseries 25/28

Definition A transserial derivation on 퐓 is a derivation ∂:퐓 →퐓 such that TD1. ∂ is strong (i.e., it preserves infinite summation); TD2. ∂ log f =∂ f / f for all f ∈퐓>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. What next? 26/28

(surreal) Numbers

H-fields Transseries

Germs (in HARDY fields) What next? 26/28

(surreal) Numbers

beyond H-fields Transseries

Germs (in HARDY fields) What next? 26/28

(surreal) Numbers

beyond H-fields Hyperseries

Germs (in HARDY fields) Hyperseries …=… Surreal numbers 27/28

Solving functional equations ↝ lack of hyperlogarithms

log휔 log x = log휔 x− 1 Hyperseries …=… Surreal numbers 27/28

Solving functional equations ↝ lack of hyperlogarithms

log휔 log x = log휔 x− 1 1 log휔 x = xlog xlog log x⋯ Hyperseries …=… Surreal numbers 27/28

Solving functional equations ↝ lack of hyperlogarithms

log휔 log x = log휔 x− 1 1 log훼 x = log훽 x 훽<훼 Hyperseries …=… Surreal numbers 27/28

Solving functional equations ↝ lack of hyperlogarithms

log휔 log x = log휔 x− 1 1 log훼 x = log훽 x 훽<훼

Problem with ∂BM

∂BM(exp휔(exp휔 휔)) = exp휔′ (exp휔 x) ≠ exp휔′ (exp휔 x)exp휔′ x Hyperseries …=… Surreal numbers 27/28

Solving functional equations ↝ lack of hyperlogarithms

log휔 log x = log휔 x− 1 1 log훼 x = log훽 x 훽<훼

Problem with ∂BM

∂BM(exp휔(exp휔 휔)) = exp휔′ (exp휔 x) ≠ exp휔′ (exp휔 x)exp휔′ x

Conjecture For a suitable definition of the class Hy of hyperseries (including the nested ones), we have No≅Hy for the 휙:Hy⟶No; f ⟼ f (휔). Thank you!

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