Surreal models of the reals with exponentiation A. Berarducci - - PowerPoint PPT Presentation

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 with Mantova, 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 exponential field is isomorphic to a truncation closed subfield of a generalized series field over R. We attempt to classify all possible logarithms on a class of truncation closed subfields and we study the question whether these exponential-logarithmic fields admit a transserial derivation, namely a strongly additive derivation of Hardy type compatible with exp.

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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 called exponential-logarithmic field. It may not be o-minimal, or a model of Texp.

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Valuations

Let K be a real closed field, let O(1) ⊆ K be a convex valuation ring with maximal 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 of Texp = Th(Rexp), and O(1) is closed under exp, one can take k to be a model of Texp [van den Dries, 1995]. Example: K = field of germs at +∞ of functions f : R → R definable in an

• -minimal expansion of R;

O(1) = germs of bounded functions;

• (1) = germs of functions tending to 0;

k = R.

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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 ≍ y (strict domination); x ∼ y if x − y ≺ x (x is asymptotic to y). We have O(1) = {x : x 1};

• (1) = {x : x ≺ 1};

x ≺ y if and only if c|x| ≤ |y| for all c ∈ O(1) (or equivalently for all c ∈ 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).

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H-fields

Let K be a real closed field. Given a derivation ∂ : K → K, let O(1) be the convex hull of ker(∂), and let o(1) be the maximal ideal of O(1). We say 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 case k = ker(∂) is the residue field. Given x, y in a H-field K with y ≍ 1, we have: x y implies ∂x ∂y; x ≍ y implies ∂x ≍ ∂y; x ≺ y implies ∂x ≺ ∂y; x ∼ y implies ∂x ∼ ∂y.

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Example: germs of definable functions

Let K be the field of germs at +∞ of functions f : R → R definable in an

• -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 makes K into a H-field with ker ∂ = R. More generally any Hardy field is an H-field, where a Hardy field is a field

• f germs of eventually C 1 functions on R closed under differentiation.
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Hahn groups

By a chain we mean a linearly ordered set. Given a chain Γ and an

• rdered abelian group (C, +, <), the Γ-sum of C, written
• ΓC,

is the abelian group of all functions f : Γ → C with reverse well-ordered support {γ ∈ Γ : f (γ) = 0} and pointwise addition, ordered by declaring f > 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

γiri, where α is an ordinal and (γi)i<α is a decreasing enumeration of the support.

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Hahn fields

Given 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. Its elements are functions f : G → k with reverse well-ordered supports, which we write either in the form f =

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

form f =

• i<α

giri 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 multiplication is given by the usual Cauchy product. We order k((G)) according to the sign of the leading coefficient, namely f > 0 ⇐ ⇒ r0 > 0. Note that the additive reduct of k((G)) is Gk.

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Summability

A family (fi)i∈I of elements of k((G)) is summable if the union of the supports 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 this case we define f =

• i∈I

fi as the unique element of k((G)) such that, for each g ∈ G, the coefficient

• f 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.

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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 giri belongs to K, then i<β giri

belongs 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 of Tan = Th(Ran) [van den Dries et al., 1994]. The same remains true replacing R with an arbitrary model k of Tan.

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Global exp

Fix a regular uncountable ordinal κ and let k((G))κ ⊆ k((G)) be the subfield consisting of the series

i<α giri whose length α is less

than κ. Then k((G))κ is an analytic subfield. If G = 1, the full Hahn field K = R((G)) never admits an exponential function R [Kuhlmann et al., 1997]. However for suitable choices of κ and G, R((G))κ does admit an exponential function [Kuhlmann and Shelah, 2005].

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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 the form

i<α giri 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 the following properties:

1 For ε ≺ 1 in K, log(1 + ε) = ∞

i=1 (−1)i+1 i

εi;

2 log(G) = K↑

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The omega-map

Let K ⊆ k((G)) be an analytic subfield, for instance K = k((G))κ. We shall call omega-map any isomorphism of ordered groups ω : (K, +, <) ∼ = (G, ·, <). The definition is ispired by Conway’s omega map on the field of surreal numbers No = R((ωNo))On [Conway, 1976, Gonshor, 1986], which extends Cantor’s normal form of an ordinal number.

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From the omega-map to the logarithm

Theorem: Suppose that k is an exponential logarithmic field, let κ be a regular uncountable cardinal, and let K = k((G))κ ⊆ k((G)). Suppose that K admits an omega map ω : K ∼ = G. Then:

1 K can be endowed with an analytic logarithm log : K>0 → K

extending the given logarithm on k.

2 Let h : K ∼

= K>0 be any chain isomorphism. There there is a unique analytic 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(ωωx) = 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 .

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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) for all 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) ≺ ωx for some x, then (K, log) is not o-minimal (as otherwise you could prove by o-minimality that exp grows faster than any polynomial). Question: Given ω : (K, +, <) ∼ = (G, ·, <), can we find h : (K, <) ∼ = (K<0, <) such that h(x) ≺ ωx for every x ∈ K?

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A multiplicative version of the Hahn group

Let k be an exponential-logarithmic field. Let κ be a regular uncountable

• cardinal. Given a chain Γ, let

(

• κ

tΓk, ·, <) ∼ = (

• κ

Γk) be a multiplicative copy of the Hahn group. Its elements can be written either in the form t

• i γiri and multiplied by adding the exponents, or as

formal products g =

• i<α

tγiri := t

• i<α γiri.

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.

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Construction of a field with an omega-map I

Let H(Γ) :=

κ tΓk and consider H as a functor from chains to groups. If

j : Γ → Γ′, then H(j) : H(Γ) → H(Γ′) acts on the exponents: H(j)(

• i<α

tγiri) =

• i<α

tj(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 chain embedding (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 we

• btain 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.

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Construction of an omega-map II

Once we have a chain Γ with a chain isomorphism η : Γ ∼ =

• κ

tΓk, we set G =

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

ω : (K, +) ∼ = (G, ·) by ω

• i<α giri =
• i<α

tη−1(gi)ri. Then fix h : K ∼ = K>0 and define log : K>0 → K as before, namely log(ωωx) = ωhx etc.

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How general is the omega-construction of the logarithm?

We defined an analytic logarithm on k((G))κ starting from two maps ω and h, in analogy with the surreal numbers. However there are fields of the form k((G))κ which admit an analytic logarithm but not an omega-map. If there is an analytic logarithm on k((G))κ, a necessary and sufficient condition for the existence of an omega-map is that there is a chain isomorphism ψ : G ∼ = G >1. Indeed, given ψ, one can define ωg = eψ(g) for g ∈ G and extend this to an omega map on the whole of K in the natural way. Viceversa, given the omega map ω : K ∼ = G, define ψ(g) = log(ωg)

• r simply observe that K ∼

= K>0 as a chain and get ωK = G ∼ = G >1 = ωK>0.

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Derivations

Let K ⊆ k((G)) be an analytic subfield with an analytic logarithm. A transserial 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 giri = i<α ∂(gi)ri;

3 ∂ex = ex∂(x).

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The S. Kuhlman-Shelah approach

More general than the omega-approach. Let κ be a regular uncountable

• cardinal. Recall the definition of the functor H(Γ) =

κ tΓk.

We have seen that starting from an initial chain Γ0 and a chain embedding ι0 : Γ0 → H(Γ0), we can iterate κ-times the functor and obtain a final chain isomorphism ι : Γ ∼ = H(Γ) extending ι0. One can do the same with H(Γ)>1 instead of H(Γ). Starting with ι0 : Γ0 → H(Γ0)>1 we obtain ι : Γ ∼ = H(Γ)>1. We then take G = H(Γ) and consider the analytic logarithm on K = k((G))κ whose restriction to G is log(

i tγiritγ) = i ι(γi)ri.

If ι(γ) < tγr for all γ ∈ Γ and positive r ∈ k, then K satisfies the growth axiom. We can show that all analytic logarithms on fields of the form k((G))κ are isomorphism to models arising in this way and call this the iota-construction.

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Models with a transserial derivations

Start with Γ0 = {γn : n ∈ Z} (ordered as Z) and a chain embedding ι0 : Γ0 → H(Γ0) =

κ tΓ0k.

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 transserial derivation (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.

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Models without a transserial derivations

If instead we define ι0 : Γ0 → H(Γ0)>1

κ

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.

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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 a chain isomorphism G ∼ = G >1, hence without an omega-map. It will however have a transserial derivation.

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Schmeling’s paths

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

• i<α girir

with gi ∈ G >1. We associate to gr a tree T(gr) as follows. The root is labeled by gr; for i < α, the ith child of the root is labeled by giri, and the descendants of giri form a subtree T(giri) defined coinductively in the same way. We call g0r0 the left-most child of gr. Let P(gr) be the set of (maximal) paths through T(gr), namely a path P is a maximal sequence P(0) = gr, P(1) = giri, . . . such that each node, with the exception of the first, is a child of the previous node.

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T(gr) is not well founded: all its paths are infinite. Now let T ′ be a well founded 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 by the 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 no bifurcations after a log-atomic. But can we always choose T ′ so that its leaves P(nP) are log-atomic?

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Log-atomic numbers

By “number” we mean “surreal number” or element of some K ⊆ k((G))κ with an analytic log.

1 Two infinite numbers x, y have the same level if there is n ∈ N such

that 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 logarithm

logn(x) is an infinite monomial.

4 ω = ω1 ∈ No is log-atomic; there is exactly one log-atomic number in

each 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].

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Variants of Schmeling’s Axiom T4

K satisfies ELT4 if for every g ∈ G and every path P ∈ T(g) there is nP such 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.

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A transserial derivation on the surreal numbers

In 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 for transserial derivation on the surreals and we define the derivative of a log-atomic surreal so as to respect the order, the log, and the above necessary 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, we ignore the paths which do not meet any log atomic. Thanks to T4− this works [Berarducci and Mantova, 2018].

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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 that logα(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,

d dx logn(x) = 1 n−1

i=0 logi(x)

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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 embedded in (No, ∂) as an ordered differential field.

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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 compatibility relations:

1 r ◦ h = r if ∂r = 0; 2 ∂f > 0 =

⇒ f ◦ x < f ◦ y whenever x < y;

3 ∂(f ◦ g) = (∂f ◦ g) · ∂g.

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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 unique composition

• : Rω × No>R → No

where Rω ⊆ No is the smallest subfield of No containing R and ω = ω1 ∈ No and closed under R, , log, exp (it is a proper class). For instance (

n logn(ω)) ◦ ( n logn(ω)) ∈ No.

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Properties

It turns out that ◦ : Rω × No>R → No is compatible with our derivation and has additional nice properties:

1 for f ∈ Rω

∂f ◦ x = lim

ε→0

f ◦ (x + ε) − f ◦ x ε

2 each f ∈ Rω is “surreal analytic”, namely:

f ◦ (x + ε) =

• n∈N

∂nf ◦ x n! · εn This suggests that No equipped with all the functions x ∈ No>R → f ◦ x for f ∈ Rω has nice model theoretic properties.

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A negative result

The derivation ∂ : No → No in [Berarducci and Mantova, 2018] cannot be 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 the fact that ker(∂) = R is a SET.

• A. Berarducci (University of Pisa)

Surreal models of the reals with exponentiation Paris, IHP, 6-8 Feb. 2018 36 / 39

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