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 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 (i.e. reverse well-ordered) formal Hahn series of real powers of the posive infinite indeterminate , exponentials, logarithms and their compositions, with real coefficients. Two important additionnal 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).
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 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 transsexponential terms, for instance formal solutions of the Abel equation .
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).
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 ).
This essentially applies the distributive law to the product; because the series is well-based, the inner sum is always finite.
- (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).
Log-exp transseries as iterated Hahn series
We first define the subfield of of so-called log-free transseries. Those are transseries which exclude any logarithmic term.
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.
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 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 (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 transseries
- 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.
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 (this is linked to the fact that those fields contain no transsexponential function).
Operations on transseries
Operations on the differential exponential ordered field
Transseries have very strong closure properties, and many operations can be defined on transseries:
- Log-exp transseries form an exponentially closed ordered field: the exponential and logarithmic functions are total. For example:
- 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 transseries
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 .
- 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.
Decidability and model theory
Theory of differential ordered valued differential field
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 field
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 . 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.
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