Transseries

Summary

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 (or in every case if using special semantics such as through infinite surreal numbers), 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[1] and Ecalle[2] 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-examplesEdit

Informally speaking, exp-log transseries are well-based (i.e. reverse well-ordered) formal Hahn series of real powers of the positive 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:

  — this series is not well-based.
  — the logarithmic depth of this series is infinite
  — the exponential and logarithmic depths of this series are infinite

It is possible to define differential fields of transseries containing the two last series; they belong respectively to   and   (see the paragraph Using surreal numbers below).

IntroductionEdit

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   (or the germ of   at infinity).

The field of transseries can be intuitively viewed as a formal generalization of 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 (also, because we care only about asymptotic behavior, pointwise convergence is not dispositive).

Because of the comparability, transseries do not include oscillatory growth rates (such as  ). 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 transexponential terms, for instance formal solutions   of the Abel equation  .[3]

Formal constructionEdit

Transseries can be defined as formal (potentially infinite) 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 (and uniquely) assigned actual growth rates (that agree with the formal operations on transseries) 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 (  is a transmonomial but is not monic unless the coefficient  ; each   is different; the order of the summands is irrelevant).

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   (see well-ordering).

A monic transmonomial is one of 1, x, log x, log log x, ..., epurely_large_transseries.

Note: Because  , we do not include it as a primitive, but many authors do; log-free transseries do not include   but   is permitted. Also, circularity in the definition is avoided because the purely_large_transseries (above) will have lower exponential depth; the definition works by recursion on the exponential depth. See "Log-exp transseries as iterated Hahn series" (below) for a construction that uses   and explicitly separates different stages.

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 (so we cannot have x + log x + log log x + ...).

Addition of transseries is termwise:   (absence of a term is equated with a zero coefficient).

Comparison:

The most significant term of   is   for the largest   (because the sum is well-based, this exists for nonzero transseries).   is positive iff the coefficient of the most significant term is positive (this is why we used 'purely large' above). X > Y iff X − Y is positive.

Comparison of monic transmonomials:

  – these are the only equalities in our construction.
 
  iff   (also  ).

Multiplication:

 
 

This essentially applies the distributive law to the product; because the series is well-based, the inner sum is always finite.

Differentiation:

 
 
 
  (division is defined using multiplication).

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   (where   is the absolute value).

Other constructionsEdit

Log-exp transseries as iterated Hahn seriesEdit

Log-free transseriesEdit

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 (i.e. reverse well-ordered) support, equipped with pointwise sum and Cauchy product (see Hahn series). In  , we distinguish the (non-unital) subring   of purely large transseries, which are series whose support contains only monomials lying strictly above  .

We start with   equipped with the product   and the order  .
If   is such that  , and thus   and   are defined, we let   denote the set of formal expressions   where   and  . This forms a linearly ordered commutative group under the product   and the lexicographic order   if and only if   or (  and  ).

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 (any of them could be zero). 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 transseriesEdit

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   (largest element of its support),   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 numbersEdit

Direct construction of log-exp transseriesEdit

One may also define the field of log-exp transseries as a subfield of the ordered field   of surreal numbers.[4] The field   is equipped with Gonshor-Kruskal's exponential and logarithm functions[5] and with its natural structure of field of well-based series under Conway normal form.[6]

Define  , the subfield of   generated by   and the simplest positive infinite surreal number   (which corresponds naturally to the ordinal  , and as a transseries to the series  ). Then, for  , define   as the field generated by  , exponentials of elements of   and logarithms of strictly positive elements of  , as well as (Hahn) 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 transseriesEdit

  • Continuing this process by transfinite induction on   beyond  , taking unions at limit ordinals, one obtains a proper class-sized field   canonically equipped with a derivation and a composition extending that of   (see Operations on transseries below).
  • If instead of   one starts with the subfield   generated by   and all finite iterates of   at  , and for   is the subfield generated by  , exponentials of elements of   and sums of summable families in  , then one obtains an isomorphic copy the field   of exponential-logarithmic transseries, which is a proper extension of   equipped with a total exponential function.[7]

The Berarducci-Mantova derivation[8] 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   (this is linked to the fact that those fields contain no transexponential function).

Additional propertiesEdit

Operations on transseriesEdit

Operations on the differential exponential ordered fieldEdit

Transseries have very strong closure properties, and many operations can be defined on transseries:

 
  • Logarithm is defined for positive arguments.
  • Log-exp transseries are real-closed.
  • Integration: every log-exp transseries   has a unique antiderivative with zero constant term  ,   and  .
  • Logarithmic antiderivative: for  , there is   with  .

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 transseriesEdit

An original property of   is that it admits a composition   (where   is the set of positive infinite log-exp transseries) 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  .

PropertiesEdit
  • Associativity: for   and  , we have   and  .
  • Compatibility of right-compositions: For  , the function   is a field automorphism of   which commutes with formal sums, sends   onto  ,   onto   and   onto  . We also have  .
  • Unicity: the composition is unique to satisfy the two previous properties.
  • Monotonicity: for  , the function   is constant or strictly monotonous on  . The monotony depends on the sign of  .
  • Chain rule: for   and  , we have  .
  • Functional inverse: for  , there is a unique series   with  .
  • Taylor expansions: each log-exp transseries   has a Taylor expansion around every point in the sense that for every   and for sufficiently small  , we have
 
where the sum is a formal Hahn sum of a summable family.
  • Fractional iteration: for   with exponentiality   and any real number  , the fractional iterate   of   is defined.[9]

Decidability and model theoryEdit

Theory of differential ordered valued differential fieldEdit

The   theory of   is decidable and can be axiomatized as follows (this is Theorem 2.2 of Aschenbrenner et al.):

  •   is an ordered valued differential field.
  •  
  •  
  •  
  •  
  • Intermediate value property (IVP):
 
where P is a differential polynomial, i.e. a polynomial in  

In this theory, exponentiation is essentially defined for functions (using differentiation) but not constants; in fact, every definable subset of   is semialgebraic.

Theory of ordered exponential fieldEdit

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

Hardy fieldsEdit

  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  . (This conjecture is informal since we have not defined which isomorphisms of Hardy fields into differential subfields of   are permitted.)   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  .[10] 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.[11]

See alsoEdit

ReferencesEdit

  1. ^ Dahn, Bernd and Göring, Peter, Notes on exponential-logarithmic terms, Fundamenta Mathematicae, 1987
  2. ^ Ecalle, Jean, Introduction aux fonctions analyzables et preuve constructive de la conjecture de Dulac, Actualités mathématiques (Paris), Hermann, 1992
  3. ^ Schmeling, Michael, Corps de transséries, PhD thesis, 2001
  4. ^ Berarducci, Alessandro and Mantova, Vincenzo, Transseries as germs of surreal functions, Transactions of the American Mathematical Society, 2017
  5. ^ Gonshor, Harry, An Introduction to the Theory of Surreal Numbers, 'Cambridge University Press', 1986
  6. ^ Conway, John, Horton, On numbers and games, Academic Press, London, 1976
  7. ^ Kuhlmann, Salma and Tressl, Marcus, Comparison of exponential-logarithmic and logarithmic-exponential series, Mathematical Logic Quarterly, 2012
  8. ^ Berarducci, Alessandro and Mantova, Vincenzo, Surreal numbers, derivations and transseries, European Mathematical Society, 2015
  9. ^ Edgar, G. A. (2010), Fractional Iteration of Series and Transseries, arXiv:1002.2378, Bibcode:2010arXiv1002.2378E
  10. ^ Aschenbrenner, Matthias, and van den Dries, Lou and van der Hoeven, Joris, On Numbers, Germs, and Transseries, In Proc. Int. Cong. of Math., vol. 1, pp. 1–24, 2018
  11. ^ Boshernitzan, Michael, Hardy fields and existence of transexponential functions, In aequationes mathematicae, vol. 30, issue 1, pp. 258–280, 1986.
  • Edgar, G. A. (2010), "Transseries for beginners", Real Analysis Exchange, 35 (2): 253–310, arXiv:0801.4877, doi:10.14321/realanalexch.35.2.0253.
  • Aschenbrenner, Matthias; Dries, Lou van den; Hoeven, Joris van der (2017), On Numbers, Germs, and Transseries, arXiv:1711.06936, Bibcode:2017arXiv171106936A.