Surreal models of the reals with exponentiationmodvac18.math.ens.fr/slides/Berarducci.pdf · The de...

Surreal models of the reals with exponentiation

A. Berarducci

University of Pisa

Paris, IHP, 6-8 Feb. 2018

A. Berarducci (University of Pisa) Surreal models of the reals with exponentiation Paris, IHP, 6-8 Feb. 2018 1 / 39

Introduction

I will report on various results on surreal numbers, exponential fields,derivations, transseries. Part of this is an ongoing collaboration withMantova, while other parts are in collaboration with S. Kuhlmann,Mantova, Matusinski. Some published results are in the bibliography.

We are interested in truncation closed subfields of generalized series fields.Examples include Ecalle’s transseries, the LE-series, the κ-bounded series,and the surreal numbers. For motivations see [Aschenbrenner et al., 2017].

Ressayre proved that every model of the theory of the real exponentialfield is isomorphic to a truncation closed subfield of a generalized seriesfield over R.

We attempt to classify all possible logarithms on a class of truncationclosed subfields and we study the question whether theseexponential-logarithmic fields admit a transserial derivation, namely astrongly additive derivation of Hardy type compatible with exp.


Exponential-logarithmic fields

Given a real closed field K, a logarithm on K is an isomorphism

log : (K>0, ·, <)→ (K,+, <)

and an exponential function is an isomorphism

exp : (K,+, <)→ (K>0, ·, <).

The inverse of an exp is a log and the inverse of a log is an exp.

If K has a log (equivalently an exp), it will be calledexponential-logarithmic field.

It may not be o-minimal, or a model of Texp.


Valuations

Let K be a real closed field, let O(1) ⊆ K be a convex valuation ring withmaximal ideal o(1). Then there is a subfield k ⊆ K such that

O(1) = k + o(1).

If K has an exponential function making it into a model ofTexp = Th(Rexp), and O(1) is closed under exp, one can take k to be amodel of Texp [van den Dries, 1995].

Example:K = field of germs at +∞ of functions f : R→ R definable in ano-minimal expansion of R;O(1) = germs of bounded functions;o(1) = germs of functions tending to 0;k = R.


Domination

For x , y ∈ K we define:

x � y if |x | ≤ c |y | for some c ∈ O(1) (domination);

x � y if x � y and y � x (comparability);

x ≺ y if x � y and x 6� y (strict domination);

x ∼ y if x − y ≺ x (x is asymptotic to y).

We have

O(1) = {x : x � 1};o(1) = {x : x ≺ 1};x ≺ y if and only if c |x | ≤ |y | for all c ∈ O(1) (or equivalently for allc ∈ k);

x � y if and only if x = cy(1 + ε) for some c ∈ k and ε ∈ o(1);

x ∼ y if and only if x = y(1 + ε) for some ε ∈ o(1).


H-fields

Let K be a real closed field. Given a derivation ∂ : K→ K, let O(1) bethe convex hull of ker(∂), and let o(1) be the maximal ideal of O(1). Wesay that ∂ is of H-type if

1 O(1) = ker(∂) + o(1);

2 for all x > ker(∂), we have ∂x > 0.

K is a H-field if it has a derivation of H-type. Notice that in in this casek = ker(∂) is the residue field.

Given x , y in a H-field K with y 6� 1, we have:

x � y implies ∂x � ∂y ;

x � y implies ∂x � ∂y ;

x ≺ y implies ∂x ≺ ∂y ;

x ∼ y implies ∂x ∼ ∂y .


Example: germs of definable functions

Let K be the field of germs at +∞ of functions f : R→ R definable in ano-minimal expansion of R.

Any such function f : R→ R is eventually of class C 1.

By differentiatin the germ of f we obtain a derivation ∂ on K which makesK into a H-field with ker ∂ = R.

More generally any Hardy field is an H-field, where a Hardy field is a fieldof germs of eventually C 1 functions on R closed under differentiation.


Hahn groupsBy a chain we mean a linearly ordered set. Given a chain Γ and anordered abelian group (C ,+, <), the Γ-sum of C , written∑

ΓC ,

is the abelian group of all functions f : Γ→ C with reverse well-orderedsupport {γ ∈ Γ : f (γ) 6= 0} and pointwise addition, ordered by declaringf > 0 if f (γ) > 0, where γ is the biggest element in the support.

We write an element of∑

ΓC in the form∑γ∈Γ

γrγ ,

representing the function sending γ ∈ Γ to rγ ∈ C , or also in the form∑i<α

γi ri ,

where α is an ordinal and (γi )i<α is a decreasing enumeration of thesupport.


Hahn fieldsGiven a field k and a multiplicative ordered abelian group G , let

k((G ))

denote the Hahn field with coefficients in k and monomials in G . Itselements are functions f : G → k with reverse well-ordered supports, whichwe write either in the form f =

∑γ∈G grg , where rg = f (g), or in in the

formf =

∑i<α

gi ri

where α is an ordinal, (gi )i<α is a decreasing enumeration of the support,and ri = f (gi ) ∈ k∗. Addition is defined componentwise and multiplicationis given by the usual Cauchy product. We order k((G )) according to thesign of the leading coefficient, namely

f > 0 ⇐⇒ r0 > 0.

Note that the additive reduct of k((G )) is∑

Gk.A. Berarducci (University of Pisa) Surreal models of the reals with exponentiation Paris, IHP, 6-8 Feb. 2018 9 / 39

SummabilityA family (fi )i∈I of elements of k((G )) is summable if the union of thesupports of the elements fi is reverse well-ordered and, for ech g ∈ G ,there are only finitely many i ∈ I such that g is in the support of fi . In thiscase we define

f =∑i∈I

fi

as the unique element of k((G )) such that, for each g ∈ G , the coefficientof g in f is the sum

∑i∈I ci ∈ k, where ci is the coefficient of g in fi . This

makes sense since only finitely many ci are non-zero.

By [Neumann, 1949] for any power series

P(x) =∑n∈N

anxn

with coefficients in k and for any ε ≺ 1 in k((G )), the family (anεn)n∈N is

summable, so we can define

P(ε) :=∑n∈N

anεn.


Analytic subfields

Let K ⊆ k((G )) be a subfield. We say that K is an analytic subfield if

1 K is truncation closed: if∑

i<α gi ri belongs to K, then∑

i<β gi ribelongs to K for every β ≤ α;

2 K contains k and G ;

3 If P(x) =∑

n∈N anxn is a power series with coefficients in k and

ε ≺ 1 is in K, then the element P(ε) =∑

n∈N anεn ∈ k((G )) lies in

the subfield K. Similarly for power series in seversal variables.

When k = R, any analytic subfield K of k((G )) is naturally a model ofTan = Th(Ran) [van den Dries et al., 1994].

The same remains true replacing R with an arbitrary model k of Tan.


Global exp

Fix a regular uncountable ordinal κ and let

k((G ))κ ⊆ k((G ))

be the subfield consisting of the series∑

i<α gi ri whose length α is lessthan κ. Then k((G ))κ is an analytic subfield.

If G 6= 1, the full Hahn field K = R((G )) never admits an exponentialfunction R [Kuhlmann et al., 1997].

However for suitable choices of κ and G , R((G ))κ does admit anexponential function [Kuhlmann and Shelah, 2005].


Analytic logarithms

Let K be an analytic subfield of k((G )). Now let ,

K↑ := k((G>1)) ∩K

be the group of the purely infinite elements, namely the elements of theform

∑i<α gi ri with gi ∈ G>1 for all i . We have:

1 K↑ is a direct complement of O(1).

2 If K has a logarithm which restricts to a logarithm on k, then log(G )is also a direct complement of O(1) (exercise).

An analytic logarithm on K is a logarithm log : K>0 → K with thefollowing properties:

1 For ε ≺ 1 in K, log(1 + ε) =∑∞

i=1(−1)i+1

i εi ;

2 log(G ) = K↑


The omega-map

Let K ⊆ k((G )) be an analytic subfield, for instance K = k((G ))κ. Weshall call omega-map any isomorphism of ordered groups

ω : (K,+, <) ∼= (G , ·, <).

The definition is ispired by Conway’s omega map on the field of surrealnumbers No = R((ωNo))On [Conway, 1976, Gonshor, 1986], which extendsCantor’s normal form of an ordinal number.


From the omega-map to the logarithmTheorem: Suppose that k is an exponential logarithmic field, let κ be aregular uncountable cardinal, and let K = k((G ))κ ⊆ k((G )). Supposethat K admits an omega map ω : K ∼= G . Then:

1 K can be endowed with an analytic logarithm log : K>0 → Kextending the given logarithm on k.

2 Let h : K ∼= K>0 be any chain isomorphism. There there is a uniqueanalytic logarithm log = logω,h on K such that, for x ∈ K,

log(ω∑

i<α ωxi ri)

=∑i<α

ωh(xi )ri .

In particular,log(ωω

x) = ωhx .

For x ∈ K>0, write x = gr(1 + ε), with g ∈ G , r ∈ k>0 and ε ≺ 1.Then

log(x) = log(r) + log(g) +∞∑n=1

(−1)n+1 εn

n.


Growth axiom

Given ω, h, we have endowed K = k((G ))κ with a logarithm, hence an exp.

We say that K satisfies the growth axiom (at infinity), if xn < exp(x) forall x > k.

This is equivalent to say that h(x) ≺ ωx for every x .

If h(x) ≺ ωx for every x , and k |= Tan,exp, then (K, log) |= Tan,exp.

If If h(x) 6≺ ωx for some x , then (K, log) is not o-minimal (as otherwise youcould prove by o-minimality that exp grows faster than any polynomial).

Question: Given ω : (K,+, <) ∼= (G , ·, <), can we findh : (K, <) ∼= (K<0, <) such that h(x) ≺ ωx for every x ∈ K?


A multiplicative version of the Hahn group

Let k be an exponential-logarithmic field. Let κ be a regular uncountablecardinal. Given a chain Γ, let

(∏κ

tΓk, ·, <) ∼= (∑κ

Γk)

be a multiplicative copy of the Hahn group. Its elements can be writteneither in the form t

∑i γi ri and multiplied by adding the exponents, or as

formal products

g =∏i<α

tγi ri := t∑

i<α γi ri .

where α < κ, (γi )i<α is a decreasing sequence in Γ and ri ∈ k∗.

We have g > 1 ⇐⇒ r0 > 0 and tγ > tβ ⇐⇒ γ > β.


Construction of a field with an omega-map ILet H(Γ) :=

∏κ t

Γk and consider H as a functor from chains to groups. Ifj : Γ→ Γ′, then H(j) : H(Γ)→ H(Γ′) acts on the exponents:

H(j)(∏i<α

tγi ri ) =∏i<α

t j(γi )ri .

Let F be the forgetful functor from ordered groups to chains.

Let Γ0 be a chain, let Γ1 = F (H(Γ0)), and let j0 : Γ0 → Γ1 be a chainembedding (for instance j0(γ) = tγ).

Now let Γ2 = F (H(Γ1)) and let j1 : Γ1 → Γ2 be F (H(j0)).

Iterate taking direct limits at limit stages. When we arrive at stage κ weobtain a chain isomorphism jκ : Γκ ∼= F (H(Γκ)) and we set η = jκ, Γ = Γκ.

We have thus constructed a chain Γ with a chain isomorphism

η : Γ ∼=∏κ

tΓk.


Construction of an omega-map II

Once we have a chain Γ with a chain isomorphism

η : Γ ∼=∏κ

tΓk,

we set G =∏κ t

Γk, K = k((G ))κ, and we define an omega-map

ω : (K,+) ∼= (G , ·)

by

ω∑

i<α gi ri =∏i<α

tη−1(gi )ri .

Then fix h : K ∼= K>0 and define log : K>0 → K as before, namelylog(ωω

x) = ωhx etc.


How general is the omega-construction of the logarithm?We defined an analytic logarithm on k((G ))κ starting from two maps ωand h, in analogy with the surreal numbers.

However there are fields of the form k((G ))κ which admit an analyticlogarithm but not an omega-map.

If there is an analytic logarithm on k((G ))κ, a necessary and sufficientcondition for the existence of an omega-map is that there is a chainisomorphism

ψ : G ∼= G>1.

Indeed, given ψ, one can define ωg = eψ(g) for g ∈ G and extend this toan omega map on the whole of K in the natural way.

Viceversa, given the omega map ω : K ∼= G , define

ψ(g) = log(ωg )

or simply observe that K ∼= K>0 as a chain and getωK = G ∼= G>1 = ωK>0

.A. Berarducci (University of Pisa) Surreal models of the reals with exponentiation Paris, IHP, 6-8 Feb. 2018 20 / 39

Derivations

Let K ⊆ k((G )) be an analytic subfield with an analytic logarithm. Atransserial derivation is a derivation ∂ : K→ K such that

1 ∂ is of H-type:ker(∂) = k, O(1) = k + o(1), x > k =⇒ ∂x > 0;

2 ∂∑

i<α gi ri =∑

i<α ∂(gi )ri ;

3 ∂ex = ex∂(x).


The S. Kuhlman-Shelah approach

More general than the omega-approach. Let κ be a regular uncountablecardinal. Recall the definition of the functor H(Γ) =

∏κ t

Γk.

We have seen that starting from an initial chain Γ0 and a chain embeddingι0 : Γ0 → H(Γ0), we can iterate κ-times the functor and obtain a finalchain isomorphism ι : Γ ∼= H(Γ) extending ι0.

One can do the same with H(Γ)>1 instead of H(Γ). Starting withι0 : Γ0 → H(Γ0)>1 we obtain ι : Γ ∼= H(Γ)>1.

We then take G = H(Γ) and consider the analytic logarithm onK = k((G ))κ whose restriction to G is log(

∏i tγi ri tγ) =

∑i ι(γi )ri .

If ι(γ) < tγr for all γ ∈ Γ and positive r ∈ k, then K satisfies the growthaxiom.

We can show that all analytic logarithms on fields of the form k((G ))κ areisomorphism to models arising in this way and call this theiota-construction.


Models with a transserial derivations

Start with Γ0 = {γn : n ∈ Z} (ordered as Z) and a chain embeddingι0 : Γ0 → H(Γ0) =

∏κ t

Γ0k.

Extend to a chain isomorphism ι : Γ ∼= H(Γ)>1 by the iota-construction.

Let G = H(Γ) and K = k((G ))κ. Then K has an analytic logarithm.

If we let ι0(γn) = tγn−1 , then K is a model of Texp and has a transserialderivation (by e.g. [Schmeling, 2001, Berarducci and Mantova, 2018]).

Moreover it contains a copy of the transseries in x > k, with

x = log(tγ0), logn(x) = tγ−n , expn(x) = tγn .


Models without a transserial derivations

If instead we define ι0 : Γ0 → H(Γ0)>1κ by the formula ι0(γn) = tγn−1tγn−2 ,

the resulting model K has no transserial derivations [BKMM, in progress].In this case we have elements αn = log(tγn) with log(αn) = αn−1 + αn−2.Thus

αn = eαn−1+αn−2 .

Too many bifurcations. This is bad for a transserial derivation.


Digression: a model without omega-map

Start with Γ0 = ω1×Z and ι0 : Γ0 → H(Γ0) given by ι0((α, n)) = t(α,n−1).

If we do the iota-construction get a model K = k((G ))κ of Texp without achain isomorphism G 6∼= G>1, hence without an omega-map.

It will however have a transserial derivation.


Schmeling’s paths

Recall that G = exp(K↑). So if g ∈ G and r ∈ k∗, we can write

gr = e∑

i<α gi ri r

with gi ∈ G>1.

We associate to gr a tree T (gr) as follows. The root is labeled by gr ; fori < α, the ith child of the root is labeled by gi ri , and the descendants ofgi ri form a subtree T (gi ri ) defined coinductively in the same way. We callg0r0 the left-most child of gr .

Let P(gr) be the set of (maximal) paths through T (gr), namely a path Pis a maximal sequence P(0) = gr ,P(1) = gi ri , . . . such that each node,with the exception of the first, is a child of the previous node.


T (gr) is not well founded: all its paths are infinite. Now let T ′ be a wellfounded subtree of P(gr) with the property that each path P in T (gr)extends a (necessarily unique) path in T ′.

Path formula: If ∂ is a transserial derivation, then ∂(gr) is determined bythe derivative of the leaves of T ′ through th following formula

∂(gr) =∑P∈T ′

∏i<nP

P(i) · ∂(P(nP))

where P = 〈P(0),P(1), . . . ,P(nP)〉.

So if you want to define a transserial derivation it suffices to define∂P(nP), and the hope is that if nP is big enough P(nP) will be a“log-atomic” element of K, making the task easier, because there are nobifurcations after a log-atomic.

But can we always choose T ′ so that its leaves P(nP) are log-atomic?


Log-atomic numbersBy “number” we mean “surreal number” or element of some K ⊆ k((G ))κwith an analytic log.

1 Two infinite numbers x , y have the same level if there is n ∈ N suchthat logn |x | ∼ logn |y |.

2 In the LE-series there are countably many levels: logn(x), x , expn(x).

3 An infinite monomial x ∈ G is log-atomic if each iterated logarithmlogn(x) is an infinite monomial.

4 ω = ω1 ∈ No is log-atomic; there is exactly one log-atomic number ineach level of No.

5 there is a proper class L = {λx : x ∈ No} of log-atomic numbers,including trans-exponential, trans-logarithmic, and “intermediate”numbers:

λ−ω < logn(ω) < ω < λ1/2 < exp(ω) < expn(ω) < λω

In general λx−1 = log(λx) [Aschenbrenner et al., 2015].


Variants of Schmeling’s Axiom T4

K satisfies ELT4 if for every g ∈ G and every path P ∈ T (g) there is nPsuch that P(nP) is log-atomic [Kuhlmann and Matusinski, 2015].

This is false in the surreals, however they satisfy the weaker property T4−:every path in T (g) is eventually right-most[Berarducci and Mantova, 2018].

It then follows that any eventually left-most path meets a log-atomic.


A transserial derivation on the surreal numbersIn any Hardy-field with log and exp, if f , g , f − g > R, then

| log(∂f )− log(∂g)| < r |f − g |

for any f , g , f − g > R.

One can prove it differentiating efr � egr . This is a necessary condition fortransserial derivation on the surreals and we define the derivative of alog-atomic surreal so as to respect the order, the log, and the abovenecessary condition.

We try to extend by the path formula

∂(gr) =∑

P∈T (gr)

∏i<nP

P(i) · ∂(P(nP))

where nP is such that P(nP) is log-atomic. If some nP does not exist, weignore the paths which do not meet any log atomic. Thanks to T4− thisworks [Berarducci and Mantova, 2018].


Explicit formulas

Let ∂ For the “simplest” transserial derivation on the surreal.

One can show that if λ is log-atomic, then

∂(λ) =

∏∞n=0 logn(ω)∏α logα(ω)

where logα(ω) := λ−α and α ranges over the ordinals such thatlogα(ω) ≥ logn(λ) for some n.

A special case of the above formula yields

∂λ−α =1∏

β<α λ−β.

Heuristic: for n ∈ N, ddx logn(x) = 1∏n−1

i=0 logi (x)


How good is the derivation?

Theorem [Aschenbrenner et al., 2015](No, ∂) is an elementary extension of the LE-series as a differential field.

Every H-field with small derivation and constant field R can be embeddedin (No, ∂) as an ordered differential field.


Axioms for a composition

1 (∑

i fi ) ◦ g =∑

i (fi ◦ g);

2 (f · g) ◦ h = (f ◦ h) · (g ◦ h);

3 exp(f ) ◦ g = exp(f ◦ g);

4 ω ◦ h = h = h ◦ ω;

5 r ◦ h = r for r ∈ R;

6 (f ◦ g) ◦ h = f ◦ (g ◦ h);

7 f < g =⇒ f ◦ x < g ◦ x ;

Given a derivation ∂ and a composition ◦ we require some compatibilityrelations:

1 r ◦ h = r if ∂r = 0;

2 ∂f > 0 =⇒ f ◦ x < f ◦ y whenever x < y ;

3 ∂(f ◦ g) = (∂f ◦ g) · ∂g .


Existence?

We cannot hope to find a composition on the whole of No:

(∑n

ω−n) ◦ 1/2 =∑n

2n =?

We may however ask whether there is a composition

◦ : No×No>R → No

We prove [Berarducci and Mantova, 2017] that there is a uniquecomposition

◦ : R〈〈ω〉〉 ×No>R → No

where R〈〈ω〉〉 ⊆ No is the smallest subfield of No containing R andω = ω1 ∈ No and closed under R,

∑, log, exp (it is a proper class).

For instance (∑

n logn(ω)) ◦ (∑

n logn(ω)) ∈ No.


Properties

It turns out that ◦ : R〈〈ω〉〉 ×No>R → No is compatible with ourderivation and has additional nice properties:

1 for f ∈ R〈〈ω〉〉

∂f ◦ x = limε→0

f ◦ (x + ε)− f ◦ xε

2 each f ∈ R〈〈ω〉〉 is “surreal analytic”, namely:

f ◦ (x + ε) =∑n∈N

∂nf ◦ xn!

· εn

This suggests that No equipped with all the functions x ∈ No>R 7→ f ◦ xfor f ∈ R〈〈ω〉〉 has nice model theoretic properties.


A negative result

The derivation ∂ : No→ No in [Berarducci and Mantova, 2018] cannotbe compatible with a composition ◦ : No×No>R → No.

Recall that ∂λω = 1∏n∈N λn

where λn = logn(ω).

Now let λ > expn(ω) for every n ∈ N.By [Berarducci and Mantova, 2018] ∂λ =

∏n logn(λ).

∂(λω ◦ λ) = (∂λω ◦ λ) · ∂λ

=

(1∏n λn◦ λ)· ∂λ

=

(1∏

n logn(λ)

)· ∂λ = 1

So there are a proper class of elements with derivative 1, contradicting thefact that ker(∂) = R is a SET.


Aschenbrenner, M., van den Dries, L., and van der Hoeven, J. (2015).The surreal numbers as a universal H-field.ArXiv 1512.02267, pages 1–17.

Aschenbrenner, M., van den Dries, L., and van der Hoeven, J. (2017).Asymptotic Differential Algebra and Model Theory of Transseries.Princeton University Press, Princeton.

Berarducci, A. and Mantova, V. (2017).Transseries as germs of surreal functions.To appear in: Transactions of the American Mathematical Society.ArXiv 1703.01995, pages 1–44.

Berarducci, A. and Mantova, V. (2018).Surreal numbers, derivations and transseries.Journal of the European Mathematical Society, 20:339–390.


Conway, J. J. H. (1976).On number and games, volume 6 of London Mathematical SocietyMonographs.Academic Press, London.

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Kuhlmann, F.-V., Kuhlmann, S., and Shelah, S. (1997).Exponentiation in power series fields.Proceedings of the American Mathematical Society,125(11):3177–3183.

Kuhlmann, S. and Matusinski, M. (2015).The Exponential-Logarithmic Equivalence Classes of Surreal Numbers.Order, 32(1):53–68.


Kuhlmann, S. and Shelah, S. (2005).κ-bounded exponential-logarithmic power series fields.Annals of Pure and Applied Logic, 136(3):284–296.

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