RAAG Annual Meeting, Passau September 5-9, 2005.

September 2, 2005

Salma Kuhlmann ^{1} Research Unit Algebra and Logic, University of Saskatchewan, McLean Hall, 106 Wiggins Road, Saskatoon, SK S7N 5E6, Canada

email: skuhlman@math.usask.ca

homepage: http://math.usask.ca/˜ skuhlman/index.html

The slides of this talk are available at: http://math.usask.ca/˜ skuhlman/slidekbd.pdf

1Partially supported by the Natural Sciences and Engineering Research Council of Canada.

κ-bounded Exponential-Logarithmic Power Series Fields.

Abstract:

• Since Wilkie’s result [9] (which established that the elementary theory T_{exp} of (ℝ, exp) is model complete and o-minimal), many o-minimal expansions of the reals have been investigated. The problem of constructing nonarchimedean models of T_{exp} (and more generally, of an ominimal expansion of the reals) gained much interest.

• In [2] it was shown that fields of generalized power series cannot admit an exponential function.

• Elaborationg on an idea of [3], we construct in [4] fields of generalized power series with support of bounded cardinality which admit an exponential.

• In this talk, we present the construction given in [4]: We give a natural definition of an exponential, which makes these fields into models of the o-minimal expansion T_{an,exp} := the theory of the reals with restricted analytic functions and exponentiation.

• We present preliminary ideas on how to introduce derivation operators on these models. The aim is to present a new class of ordered differential fields, with many interesting properties.

References:

[1] Kuhlmann, F.- V. - Kuhlmann, S.: Explicit construction of exponential-logarithmic power series, Prépublications de Paris 7 No 61, Séminaire Structures Algébriques Ordonnées, Séminaires 1995-1996.

[2] Kuhlmann, F.-V. - Kuhlmann, S. - Shelah: Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 (1997), 3177-3183 .

[3] Kuhlmann, F.-V. - Kuhlmann, S.: The exponential rank of nonarchimedean exponential fields, in: Delzell & Madden (eds): Real Algebraic Geometry and Ordered Structures, Contemp. Math. 253, AMS (2000), 181-201 .

[4] Kuhlmann, S. - Shelah: κ–bounded Exponential Logarithmic Power Series Fields, to appear in Annals for Pure and Applied Logic.

[5] Kuhlmann, S.- Tressl: A Note on Logarithmic - Exponential and Exponential - Logarithmic Power Series Fields, work in progress (2005)

[6] Schmeling: Corps de Transséries, Dissertation (2001).

[7] van den Dries - Macintyre - Marker: Logarithmic-Exponential series, Annals for Pure and Applied Logic 111 (2001), 61–113

[8] van der Hoeven: Transseries and real differential algebra, Prépublications Université Paris-Sud (2004)

[9] Wilkie: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), 1051–1094

Notations and Preliminaries.

The natural valuation.

• Let G be a totally ordered abelian group. The archimedean equivalence relation on G is defined as follows. For 0 ≠ x, 0 ≠ y ∈ G:

x ∼^{+} y if ∃n ∈ ℕ s.t. n|x| ≥ |y| and n|y| ≥ |x|

where |x| := max{x, −x}. We set x << y if for all n ∈ ℕ, n|x| < |y|. We denote by [x] is the archimedean equivalence class of x. We totally order the set of archimedean classes as follows: [y] < [x] if x << y.

• Let (K, +, ·, 0, 1, <) be an ordered field. Using the archimedean equivalence relation on the ordered abelian group (K, +, 0, <), we can endow K with the natural valuation v:

for x, y ∈ K, x, y ≠ 0 define v(x) := [x] and [x]+[y] := [xy].

Notation: Value group: v(K) := {v(x) | x ∈ K, x ≠ 0}. Valuation ring:, R_{v} := {x | x ∈ K and v(x) ≥ 0}. Valuation ideal: I_{v} := {x | x ∈ K and v(x) > 0}. Group of positive units: U_{v}^{>0} := {x | x ∈ R_{v}, x > 0, v(x) = 0}.

Ordered Exponential Fields.

An ordered field K is an exponential field if there exists a map

exp : (K, +, 0, <) ⟶ (K^{>0}, ·, 1, <)

such that exp is an isomorphism of ordered groups. A map exp with these properties will be called an exponential on K. A logarithm on K is the compositional inverse log = exp^{−1} of an exponential. WLOG, we require the exponentials (logarithms) to be v-compatible:

exp(R_{v}) = U_{v}^{>0} or log(U_{v}^{>0}) = R_{v} > .

We are interested in exponentials (logarithms) satisfying the growth axiom scheme: (GA):

∀n ∈ ℕ : x > log(x^{n}) = nlog(x) for all x ∈ K^{>0} \ R_{v} .

Via the natural valuation v, this is equivalent to

v(x) < v(log(x)) for all x ∈ K^{>0} \ R_{v} .  (1)

A logarithm log is a (GA)-logarithm if it satisfies (1).

Hahn Groups and Fields.

• Let Γ be any totally ordered set and R any ordered abelian group. Then R^{Γ} is the set of all maps g from Γ to R such that the support {γ ∈ Γ | g(γ) ≠ 0} of g is well-ordered in Γ. Endowed with the lexicographic order and pointwise addition, R^{Γ} is an ordered abelian group, called the Hahn group.

• Representation for the elements of Hahn groups: Fix a strictly positive element 1 ∈ R (if R is a field, we take 1 to be the neutral element for multiplication). For every γ ∈ Γ, we will denote by 1_{γ} the map which sends γ to 1 and every other element to 0 (1_{γ} is the characteristic function of the singleton {γ}.) For g ∈ R^{Γ} write

g = _{γ∈Γ}^{∑} g_{γ}1_{γ}

(where g_{γ} := g(γ) ∈ R).

• For G ≠ 0 an ordered abelian group, k an archimedean ordered field, k((G)) is the (generalized) power series field with coefficients in k and exponents in G. As an ordered abelian group, this is just the Hahn group k^{G}. A series s ∈ k((G)) is written

s = _{g∈G}^{∑} s_{g}t^{g}

with s_{g} ∈ k and well-ordered support {g ∈ G | s_{g} ≠ 0}.

• The natural valuation on k((G)) is v(s) = min support s for any series s ∈ k((G)). The value group is G and the residue field is k. The valuation ring k((G^{≥0})) consists of the series with non-negative exponents, and the valuation ideal k((G^{>0})) of the series with positive exponents. The constant term of a series s is the coefficient s_{0}. The units of k((G^{≥0})) are the series in k((G^{≥0})) with a nonzero constant term.

• Additive Decomposition Given s ∈ k((G)), we can truncate it at its constant term and write it as the sum of two series, one with strictly negative exponents, and the other with non-negative exponents. Thus a complement in (k((G)), +) to the valuation ring is the Hahn group k^{G^{<0}}. We call it the canonical complement to the valuation ring and denote it by k((G^{<0})).

• Multiplicative Decomposition Given s ∈ k((G))^{>0}, we can factor out the monomial of smallest exponent g ∈ G and write s = t^{g}u with u a unit with a positive constant term. Thus a complement in (k((G))^{>0}, ·) to the subgroup U_{v}^{>0} of positive units is the group consisting of the (monic) monomials t^{g}. We call it the canonical complement to the positive units and denote it by Mon k((G)). κ–bounded Hahn Groups and Fields. Fix a regular uncountable cardinal κ.

• The κ-bounded Hahn group (R^{Γ})_{κ} ⊆ R^{Γ} consists of all maps of which support has cardinality < κ.

• The κ-bounded power series field k((G))_{κ} ⊆ k((G)) consists of all series of which support has cardinality < κ. It is a valued subfield of k((G)). We denote by k((G^{≥0}))_{κ} its valuation ring. Note that k((G))_{κ} contains the monic monomials. We denote by k((G^{<0}))_{κ} the complement to k((G^{≥0}))_{κ}.

• Our first goal is to define an exponential (logarithm) on k((G))_{κ} (for appropriate choice of G). From the above discussion, we get: Proposition 0.1 Set K = k((G))_{κ}. Then (K, +, 0, <) decomposes lexicographically as the sum:

(K, +, 0, <) = k((G^{<0}))_{κ} ⊕ k((G^{≥0}))_{κ} .  (2)

(K^{>0}, ·, 1, <) decomposes lexicographically as the product:

(K^{>0}, ·, 1, <) = Mon (K) × U_{v}^{>0}  (3)

Moreover, Mon (K) is order isomorphic to G through the isomorphism t^{g} ↦ −g.

Proposition 0.1 allows us to achieve our goal in two main steps; by defining the logarithm on Mon (K) and on U_{v}^{>0}.

The Main Theorem

Theorem 0.2 Let Γ be a chain, G = (ℝ^{Γ})_{κ} and K = R((G))_{κ}. Assume that

l : Γ → G^{<0}

is an embedding of chains. Then l induces an embedding of ordered groups (a prelogarithm)

log : (K^{>0}, ·, 1, <) ⟶ (K, +, 0, <)

as follows: given a ∈ K^{>0}, write a = t^{g}r(1 + ε) (with g = ∑_{γ∈Γ} g_{γ}1_{γ}, r ∈ ℝ^{>0}, ε infinitesimal), and set

log(a) := − _{γ∈Γ}^{∑} g_{γ}t^{l(γ)} + log r + ∞ ; ∑ ; i=1(−1)^{(i−1)} \frac{ε^{i}}{i} (4)

This prelogarithm satisfies

v(log t^{g}) = l(min support g)  (5)

Moreover, log is surjective (a logarithm) if and only if l is surjective, and log satisfies GA if and only if

l(min support g) > g  for all g ∈ G^{<0} . (6)

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Prelogarithmic fields of power series.

Example 0.3 Power Series fields endowed with a basic prelogaritm: Let Γ be any chain, G = (ℝ^{Γ})_{κ} and K = ℝ((G))_{κ}. Then

ι : Γ → G^{<0} defined by γ ↦ −1_{γ}

is an embedding of chains, and gives rise to prelogaritm on K. However, this prelogarithm is neither surjective nor does it satisfy GA.

• To get a prelogarithm satisfying GA, we choose σ ∈ Aut (Γ) with the property that

σ(γ) > γ for all γ ∈ Γ  (7)

(We say that σ is an increasing automorphism). We set l = ι ∘ σ. Now

l : Γ → G^{<0} defined by γ ↦ −1_{σ}(γ)

is an embedding of chains satisfying (6), so gives rise to a prelogaritm on K satisfying GA.

We call (K, log) the prelogarithmic field of κ-bounded power series over (Γ, σ).

• To get a surjective prelogarithm, we have to modify Γ as in the next section.

The κ-th iterated lexicographic power of a chain.

Proposition 0.4 Let Γ ≠ ∅ be a given chain. There is a canonically constructed chain Γ_{κ} ⊇ Γ together with an isomorphism of ordered chains

ι_{κ} : Γ_{κ} → G_{κ}^{<0}

where G_{κ} := (ℝ^{Γ_{κ}})_{κ}. Moreover, every increasing σ ∈ Aut (Γ) extends canonically to an increasing σ_{κ} ∈ Aut (Γ_{κ})

We call the pair (Γ_{κ}, ι_{κ}) the κ-th iterated lexicographic power of Γ.

We are now ready to summarize the procedure of constructing the Exponential-Logarithmic field of κbounded series over (Γ, σ). Let Γ be given and σ an increasing automorphism.

• Construct Γ_{κ}, G_{κ}, ι_{κ}, and σ_{κ}.

• Set K := ℝ((G_{κ}))_{κ} and l := ι_{κ} ∘ σ_{κ}. Note that l is surjective and satisfies (6).

• Denote by log the surjective GA logarithm induced on K^{>0} by l and set exp = log^{−1}.

• (K, exp) is a model of T_{an,exp}.

Growth Rates.

• Let Γ be a chain and σ ∈ Aut (Γ) an increasing automorphism. By induction, we define the n-th iterate of σ: σ^{1}(γ) := σ(γ) and σ^{n+1}(γ) := σ(σ^{n}(γ)). Define an equivalence relation on Γ as follows: For γ, γ′ ∈ Γ, set

γ ∼_{σ} γ′ iff ∃n ∈ ℕ s.t. σ^{n}(γ) ≥ γ′ and σ^{n}(γ′) ≥ γ .

The equivalence classes [γ]_{σ} of ∼_{σ} are convex and closed under application of σ (they are the convex hulls of the orbits of σ). The order of Γ induces an order on Γ/∼_{σ}. The order type of Γ/∼_{σ} is the rank of (Γ, σ).

Example 0.5 Let Γ = ℤ ⨿ ℤ (i.e. the lexicographically ordered Cartesian product ℤ × ℤ) endowed with the automorphism σ((x, y)) := (x, y + 1). The rank of σ is ℤ. Now consider the increasing automorphism τ((x, y)) := (x + 1, y). The rank of τ is 1.

• Let K be a real closed field and log a (GA)- logarithm on K^{>0}. Define an equivalence relation on K^{>0} \ R_{v}:

a ∼_{log} a′ iff ∃n ∈ ℕ s.t. log_{n}(a) ≤ (a′) and log_{n}(a′) ≤ a

(where log_{n} is the n-th iterate of the log). The order type of the chain of equivalence classes is the logarithmic rank of (K^{>0}, log).

We can compute the logarithmic rank of the Exponential-Logarithmic field of κ-bounded series over (Γ, σ):

Theorem 0.6 The logarithmic rank of (ℝ((G_{κ}))_{κ}^{>0}, log) is equal to the rank of (Γ, σ).

This proof (as many other proofs) is based on the observation that every series is log-equivalent to a fundamental monomial, that is a monomial of the form

t^{−1_{γ}} with γ ∈ Γ .

Next one observes that

for all γ, γ′ ∈ Γ : t^{−1_{γ}} ∼_{log} t^{−1_{γ′}} if and only if γ ∼_{σ} γ′ .

This in turn is based on the following useful formula for log_{n}(t^{−1_{γ}}): by induction,

log _{n}(t^{−1_{γ}}) = t^{−1_{σ^{n}(γ)}} .

Remark 0.7 If Γ admits automorphisms of distinct rank, then (ℝ((G_{κ})) admits logarithms of distinct logarithmic rank. We can also use this observation to introduce transexponentials, as illustrated in the next example.

Example 0.8 Let Γ = ℤ ⨿ ℤ, σ((x, y)) := (x, y + 1), (K, log) the corresponding κ-bounded model. For the automorphism τ((x, y)) := (x + 1, y) , let L, respectively T := L^{−1} be the corresponding induced logarithm and exponential on K. Effect of σ, τ on the fundamental monomials: let γ = (x, y) ∈ Γ, then

log(t^{−1_{γ}}) = t^{−1_{σ(γ)}} ,

Whereas

L(t^{−1_{γ}}) = t^{−1_{τ(γ)}} ,

We see that, for any fundamental monomial X := t^{−1_{γ}} and any n ∈ ℕ we have:

L(X) < log_{n}(X) .

Also, a simple computation (using the fact that σ and τ commute) shows that also, for all n ∈ ℕ:

T (X) > exp_{n}(X) .

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In the next section, we see how the logarithm determines the derivation. We expect to obtain fields equipped with several distinct derivations.

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Introducing Operators.

Main Motivation: We want a “Kaplansky embedding Theorem” for ordered differential fields. The κ-bounded fields of power series are good candidates as “universal domains”. But for this to make sense, we need first to endow them with a good differential structure.

Main project: Given (Γ, σ), introduce, if possible, derivation and composition operators on Exponential-Logarithmic field of κ-bounded series over (Γ, σ).

It seems to be enough to focus on the following

Main subproject: Given (Γ, σ), introduce, if possible, derivation and composition operators on the prelogarithmic field of κ-bounded series over (Γ, σ).

Indeed, in [6] a method is developed showing the following: given derivation and composition operators (satisfying some good properties) on a “field of transseries” T, one can extend these operators to the “exponential closure”T^{exp}. It seems that this method may be adapted to our context: given derivation and composition operators (satisfying some good properties) on the prelogarithmic field of κ-bounded series over (Γ, σ), one can extend these operators to the Exponential-Logarithmic field of κ-bounded series over (Γ, σ).

Final Goal Find necessary and sufficient conditions on (Γ, σ) so that the corresponding prelogarithmic field of κbounded series (and the corresponding Exponential-Logarithmic field of κ-bounded series) admit a surjective derivation.

Derivations: We want to endow the prelogarithmic field of κ-bounded series over (Γ, σ) with a derivation D satisfying the following properties:

• D is strongly linear, that is

D _{g}^{∑} r_{g}t^{g} = _{g}^{∑} r_{g}Dt^{g} .  (8)

• D satisfies Leibniz rule:

D(ab) = aD(b) + D(a)b  (9)

• D satisfies the rule for the logarithmic derivative for a > 0:

D log a = Da/a  (10)

Reductions: The above rules direct us to perform a number of steps in trying to define derivatives:

(i) From (8) and (9), it is clear that we only need to determine Dt^{g}, for g ∈ G^{<0}. (ii) From (10) determining Dt^{g} reduces to determining D log t^{g}. (iii) By definition of log, this in turn reduces to determining D log t^{−1_{γ}}, for a fundamental monomial t^{−1_{γ}} with γ ∈ Γ. (iv) Applying (10) again we see that for any γ ∈ Γ_{0} we have:

Dt^{−1_{σ(γ)}} = t^{1_{γ}}Dt^{−1_{γ}} .

(v) Finally from (iv), we see that we only need to define Dt^{−1_{γ_{0}}} for a fixed representative γ_{0} ∈ Γ of an orbit of σ in Γ.

Example 0.9 Let Γ = ℤ endowed with the automorphism σ(z) := z + 1. For simplicity, let us choose γ_{0} = 0 and set

T := t^{−1_{0}} and DT = 1 .

Then t^{−1_{n}} = log_{n} T if n > 0, and t^{−1_{n}} = exp_{−n} T if n < 0. Therefore, for n > 0

Dt^{−1_{n}} = n−1 ; ∏ ; k=0 t^{1_{k}} and Dt^{−1_{−n}} = n ; ∏ ; k=1 t^{−1_{−k}} .

It is non-trivial to verify that these definitions induce a well-defined derivative! Example 0.10 Let Γ = ℤ ⨿ ℤ endowed with the automorphism σ((x, y)) := (x, y + 1). The rank of σ is ℤ. For each orbit of σ_{0} we fix a representative z ∈ ℤ. We set T_{z} := t^{−1_{z}}. Then {T_{z} ; z ∈ ℤ} will represent infinitely many algebraically independent variables, which will determine an infinite family {δ_{z}} of commuting partial derivatives. What about a derivation induced by the automorphism τ((x, y)) := (x + 1, y) of rank one? This is more challenging. We have countably many distinct orbits but with a single common convex hull. This suggests defining “ arbitrary iterates” log_{γ} T of the log, to capture the derivative of every fundamental monomial.

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