Transseries


In mathematics, the field of logarithmic-exponential transseries is a non-Archimedean ordered differential field which extends comparability of asymptotic growth rates of elementary nontrigonometric functions to a much broader class of objects. Each log-exp transseries represents a formal asymptotic behavior, and it can be manipulated formally, and when it converges, corresponds to actual behavior. Transseries can also be convenient for representing functions. Through their inclusion of exponentiation and logarithms, transseries are a strong generalization of the power series at infinity and other similar asymptotic expansions.
The field was introduced independently by Dahn-Göring and Ecalle in the respective contexts of model theory or exponential fields and of the study of analytic singularity and proof by Ecalle of the Dulac conjectures. It constitutes a formal object, extending the field of exp-log functions of Hardy and the field of accelerando-summable series of Ecalle.
The field enjoys a rich structure: an ordered field with a notion of generalized series and sums, with a compatible derivation with distinguished antiderivation, compatible exponential and logarithm functions and a notion of formal composition of series.

Examples and counter-examples

Informally speaking, exp-log transseries are well-based formal Hahn series of real powers of the posive infinite indeterminate, exponentials, logarithms and their compositions, with real coefficients. Two important additional conditions are that the exponential and logarithmic depth of an exp-log transseries that is the maximal numbers of iterations of exp and log occurring in must be finite.
The following formal series are log-exp transseries:
The following formal series are not log-exp transseries:
It is possible to define differential fields of transseries containing the two last series, they belong respectively to and .

Introduction

A remarkable fact is that asymptotic growth rates of elementary nontrigonometric functions and even all functions definable in the model theoretic structure of the ordered exponential field of real numbers are all comparable:
For all such and, we have or, where means. The equivalence class of under the relation is the asymptotic behavior of, also called the germ of .
The field of transseries can be intuitively viewed as a formal generalization these growth rates: In addition to the elementary operations, transseries are closed under "limits" for appropriate sequences with bounded exponential and logarithmic depth. However, a complication is that growth rates are non-Archimedean and hence do not have the least upper bound property. We can address this by associating a sequence with the least upper bound of minimal complexity, analogously to construction of surreal numbers. For example, is associated with rather than because decays too quickly, and if we identify fast decay with complexity, it has greater complexity than necessary.
Because of the comparability, transseries do not include oscillatory growth rates. On the other hand, there are transseries such as that do not directly correspond to convergent series or real valued functions. Another limitation of transseries is that each of them is bounded by a tower of exponentials, i.e. a finite iteration of, thereby excluding tetration and other transexponential functions, i.e. functions which grow faster than any tower of exponentials. There are ways to construct fields of generalized transseries including formal transsexponential terms, for instance formal solutions of the Abel equation.

Formal construction

Transseries can be defined as formal expressions, with rules defining which expressions are valid, comparison of transseries, arithmetic operations, and even differentiation. Appropriate transseries can then be assigned to corresponding functions or germs, but there are subtleties involving convergence. Even transseries that diverge can often be meaningfully assigned actual growth rates using accelero-summation, which is a generalization of Borel summation.
Transseries can be formalized in several equivalent ways; we use one of the simplest ones here.
A transseries is a well-based sum,
with finite exponential depth, where each is a nonzero real number and is a monic transmonomial.
The sum might be infinite or transfinite; it is usually written in the order of decreasing.
Here, well-based means that there is no infinite ascending sequence .
A monic transmonomial is one of 1, x, log x, log log x,..., epurely_large_transseries.
A purely large transseries is a nonempty transseries with every.
Transseries have finite exponential depth, where each level of nesting of e or log increases depth by 1.
Addition of transseries is termwise: .
Comparison:
The most significant term of is for the largest . is positive iff the coefficient of the most significant term is positive. X > Y iff X − Y is positive.
Comparison of monic transmonomials:
Multiplication:
This essentially applies the distributive law to the product; because the series is well-based, the inner sum is always finite.
Differentiation:
With these definitions, transseries is an ordered differential field. Transseries is also a valued field, with the valuation given by the leading monic transmonomial, and the corresponding asymptotic relation defined for by if .

Other constructions

Log-exp transseries as iterated Hahn series

Log-free transseries

We first define the subfield of of so-called log-free transseries. Those are transseries which exclude any logarithmic term.
Inductive definition:
For we will define a linearly ordered multiplicative group of monomials. We then let denote the field of well-based series. This is the set of maps with well-based support, equipped with pointwise sum and Cauchy product. In, we distinguish the subring of purely large transseries, which are series whose support contains only monomials lying strictly above.
The natural inclusion of into given by identifying and inductively provides a natural embedding of into, and thus a natural embedding of into. We may then define the linearly ordered commutative group and the ordered field which is the field of log-free transseries.
The field is a proper subfield of the field of well-based series with real coefficients and monomials in. Indeed, every series in has a bounded exponential depth, i.e. the least positive integer such that, whereas the series
has no such bound.
Exponentiation on :
The field of log-free transseries is equipped with an exponential function which is a specific morphism. Let be a log-free transseries and let be the exponential depth of, so. Write as the sum in where, is a real number and is infinitesimal. Then the formal Hahn sum
converges in, and we define where is the value of the real exponential function at.
Right-composition with :
A right composition with the series can be defined by induction on the exponential depth by
with. It follows inductively that monomials are preserved by so at each inductive step the sums are well-based and thus well defined.

Log-exp transseries

Definition:
The function defined above is not onto so the logarithm is only partially defined on : for instance the series has no logarithm. Moreover, every positive infinite log-free transseries is greater than some positive power of. In order to move from to, one can simply "plug" into the variable of series formal iterated logarithms which will behave like the formal reciprocal of the -fold iterated exponential term denoted.
For let denote the set of formal expressions where. We turn this into an ordered group by defining, and defining when. We define. If and we embed into by identifying an element with the term
We then obtain as the directed union
On the right-composition with is naturally defined by
Exponential and logarithm:
Exponentiation can be defined on in a similar way as for log-free transseries, but here also has a reciprocal on. Indeed, for a strictly positive series, write where is the dominant monomial of , is the corresponding positive real coefficient, and is infinitesimal. The formal Hahn sum
converges in. Write where itself has the form where and. We define. We finally set

Using surreal numbers

Direct construction of log-exp transseries

One may also define the field of log-exp transseries as a subfield of the ordered field of surreal numbers. The field is equipped with Gonshor-Kruskal's exponential and logarithm functions and with its natural structure of field of well-based series under Conway normal form.
Define, the subfield of generated by and the simplest positive infinite surreal number . Then, for, define as the field generated by, exponentials of elements of and logarithms of strictly positive elements of, as well as sums of summable families in. The union is naturally isomorphic to. In fact, there is a unique such isomorphism which sends to and commutes with exponentiation and sums of summable families in lying in.

Other fields of transseries

The Berarducci-Mantova derivation on coincides on with its natural derivation, and is unique to satisfy compatibility relations with the exponential ordered field structure and generalized series field structure of and
Contrary to the derivation in and is not surjective: for instance the series
doesn't have an antiderivative in or .

Additional properties

Operations on transseries

Operations on the differential exponential ordered field

Transseries have very strong closure properties, and many operations can be defined on transseries:
Note 1. The last two properties mean that is Liouville closed.
Note 2. Just like an elementary nontrigonometric function, each positive infinite transseries has integral exponentiality, even in this strong sense:
The number is unique, it is called the exponentiality of.

Composition of transseries

An original property of is that it admits a composition which enables us to see each log-exp transseries as a function on. Informally speaking, for and, the series is obtained by replacing each occurrence of the variable in by.
Properties

Theory of differential ordered valued differential field

The theory of is decidable and can be axiomatized as follows :
In this theory, exponentiation is essentially defined for functions but not constants; in fact, every definable subset of is semialgebraic.

Theory of ordered exponential field

The theory of is that of the exponential real ordered exponential field, which is model complete by Wilkie's theorem.

Hardy fields

is the field of accelero-summable transseries, and using accelero-summation, we have the corresponding Hardy field, which is conjectured to be the maximal Hardy field corresponding to a subfield of. is conjectured to satisfy the above axioms of. Without defining accelero-summation, we note that when operations on convergent transseries produce a divergent one while the same operations on the corresponding germs produce a valid germ, we can then associate the divergent transseries with that germ.
A Hardy field is said maximal if it is properly contained in no Hardy field. By an application of Zorn's lemma, every Hardy field is contained in a maximal Hardy field. It is conjectured that all maximal Hardy fields are elementary equivalent as differential fields, and indeed have the same first order theory as. Logarithmic-transseries do not themselves correspond to a maximal Hardy field for not every transseries corresponds to a real function, and maximal Hardy fields always contain transsexponential functions.