Transseries

We first define the subfield ${\displaystyle \mathbb {T} ^{E}}$ of ${\displaystyle \mathbb {T} ^{LE}}$ of so-called log-free transseries. Those are transseries which exclude any logarithmic term.

Inductive definition:

For ${\displaystyle n\in \mathbb {N} ,}$ we will define a linearly ordered multiplicative group of monomials ${\displaystyle {\mathfrak {M}}_{n}}$. We then let ${\displaystyle \mathbb {T} _{n}^{E}}$ denote the field of well-based series ${\displaystyle \mathbb {R} [[{\mathfrak {M}}_{n}]]}$. This is the set of maps ${\displaystyle \mathbb {R} \to {\mathfrak {M}}_{n}}$ with well-based (i.e. reverse well-ordered) support, equipped with pointwise sum and Cauchy product (see Hahn series). In ${\displaystyle \mathbb {T} _{n}^{E}}$, we distinguish the (non-unital) subring ${\displaystyle \mathbb {T} _{n,\succ }^{E}}$ of purely large transseries, which are series whose support contains only monomials lying strictly above ${\displaystyle 1}$.

We start with ${\displaystyle {\mathfrak {M}}_{0}=x^{\mathbb {R} }}$ equipped with the product ${\displaystyle x^{a}x^{b}:=x^{a+b}}$ and the order ${\displaystyle x^{a}\prec x^{b}\leftrightarrow a<b}$.

If ${\displaystyle n\in \mathbb {N} }$ is such that ${\displaystyle {\mathfrak {M}}_{n}}$, and thus ${\displaystyle \mathbb {T} _{n}^{E}}$ and ${\displaystyle \mathbb {T} _{n,\succ }^{E}}$ are defined, we let ${\displaystyle {\mathfrak {M}}_{n+1}}$ denote the set of formal expressions ${\displaystyle x^{a}e^{\theta }}$ where ${\displaystyle a\in \mathbb {R} }$ and ${\displaystyle \theta \in \mathbb {T} _{n,\succ }^{E}}$. This forms a linearly ordered commutative group under the product ${\displaystyle (x^{a}e^{\theta })(x^{a'}e^{\theta '})=(x^{a+a'})e^{\theta +\theta '}}$ and the lexicographic order ${\displaystyle x^{a}e^{\theta }\prec x^{a'}e^{\theta '}}$ if and only if ${\displaystyle \theta <\theta '}$ or (${\displaystyle \theta =\theta '}$ and ${\displaystyle a<a'}$).

The natural inclusion of ${\displaystyle {\mathfrak {M}}_{0}}$ into ${\displaystyle {\mathfrak {M}}_{1}}$ given by identifying ${\displaystyle x^{a}}$ and ${\displaystyle x^{a}e^{0}}$ inductively provides a natural embedding of ${\displaystyle {\mathfrak {M}}_{n}}$ into ${\displaystyle {\mathfrak {M}}_{n+1}}$, and thus a natural embedding of ${\displaystyle \mathbb {T} _{n}^{E}}$ into ${\displaystyle \mathbb {T} _{n+1}^{E}}$. We may then define the linearly ordered commutative group ${\textstyle {\mathfrak {M}}=\bigcup _{n\in \mathbb {N} }{\mathfrak {M}}_{n}}$ and the ordered field ${\textstyle \mathbb {T} ^{E}=\bigcup _{n\in \mathbb {N} }\mathbb {T} _{n}^{E}}$ which is the field of log-free transseries.

The field ${\displaystyle \mathbb {T} ^{E}}$ is a proper subfield of the field ${\displaystyle \mathbb {R} [[{\mathfrak {M}}]]}$ of well-based series with real coefficients and monomials in ${\displaystyle {\mathfrak {M}}}$. Indeed, every series ${\displaystyle f}$ in ${\displaystyle \mathbb {T} ^{E}}$ has a bounded exponential depth, i.e. the least positive integer ${\displaystyle n}$ such that ${\displaystyle f\in \mathbb {T} _{n}^{E}}$, whereas the series

${\displaystyle e^{-x}+e^{-e^{x}}+e^{-e^{e^{x}}}+\cdots \in \mathbb {R} [[{\mathfrak {M}}]]}$

has no such bound.

Exponentiation on ${\displaystyle \mathbb {T} ^{E}}$:

The field of log-free transseries is equipped with an exponential function which is a specific morphism ${\displaystyle \exp$ :(\mathbb {T} ^{E},+)\to (\mathbb {T} ^{E,>},\times )} . Let ${\displaystyle f}$ be a log-free transseries and let ${\displaystyle n\in \mathbb {N} }$ be the exponential depth of ${\displaystyle f}$, so ${\displaystyle f\in \mathbb {T} _{n}^{E}}$. Write ${\displaystyle f}$ as the sum ${\displaystyle f=\theta +r+\varepsilon }$ in ${\displaystyle \mathbb {T} _{n}^{E},}$ where ${\displaystyle \theta \in \mathbb {T} _{n,\succ }^{E}}$, ${\displaystyle r}$ is a real number and ${\displaystyle \varepsilon }$ is infinitesimal (any of them could be zero). Then the formal Hahn sum

${\displaystyle E(\varepsilon ):=\sum _{k\in \mathbb {N} }{\frac {\varepsilon ^{k}}{k!}}}$

converges in ${\displaystyle \mathbb {T} _{n}^{E}}$, and we define ${\displaystyle \exp(f)=e^{\theta }\exp(r)E(\varepsilon )\in \mathbb {T} _{n+1}^{E}}$ where ${\displaystyle \exp(r)}$ is the value of the real exponential function at ${\displaystyle r}$.

Right-composition with ${\displaystyle e^{x}}$:

A right composition ${\displaystyle \circ _{e^{x}}}$ with the series ${\displaystyle e^{x}}$ can be defined by induction on the exponential depth by

${\displaystyle \left(\sum f_{\mathfrak {m}}{\mathfrak {m}}\right)\circ e^{x}:=\sum f_{\mathfrak {m}}({\mathfrak {m}}\circ e^{x}),}$

with ${\displaystyle x^{r}\circ e^{x}:=e^{rx}}$. It follows inductively that monomials are preserved by ${\displaystyle \circ _{e^{x}},}$ so at each inductive step the sums are well-based and thus well defined.

Definition:

The function ${\displaystyle \exp }$ defined above is not onto ${\displaystyle \mathbb {T} ^{E,>}}$ so the logarithm is only partially defined on ${\displaystyle \mathbb {T} ^{E}}$: for instance the series ${\displaystyle x}$ has no logarithm. Moreover, every positive infinite log-free transseries is greater than some positive power of ${\displaystyle x}$. In order to move from ${\displaystyle \mathbb {T} ^{E}}$ to ${\displaystyle \mathbb {T} ^{LE}}$, one can simply "plug" into the variable ${\displaystyle x}$ of series formal iterated logarithms ${\displaystyle \ell _{n},n\in \mathbb {N} }$ which will behave like the formal reciprocal of the ${\displaystyle n}$-fold iterated exponential term denoted ${\displaystyle e_{n}}$.

For ${\displaystyle m,n\in \mathbb {N} ,}$ let ${\displaystyle {\mathfrak {M}}_{m,n}}$ denote the set of formal expressions ${\displaystyle {\mathfrak {u}}\circ \ell _{n}}$ where ${\displaystyle {\mathfrak {u}}\in {\mathfrak {M}}_{m}}$. We turn this into an ordered group by defining ${\displaystyle ({\mathfrak {u}}\circ \ell _{n})({\mathfrak {v}}\circ \ell _{n}(x)):=({\mathfrak {u}}{\mathfrak {v}})\circ \ell _{n}}$, and defining ${\displaystyle {\mathfrak {u}}\circ \ell _{n}\prec {\mathfrak {v}}\circ \ell _{n}}$ when ${\displaystyle {\mathfrak {u}}\prec {\mathfrak {v}}}$. We define ${\displaystyle \mathbb {T} _{m,n}^{LE}:=\mathbb {R} [[{\mathfrak {M}}_{m,n}]]}$. If ${\displaystyle n'>n}$ and ${\displaystyle m'\geq m+(n'-n),}$ we embed ${\displaystyle {\mathfrak {M}}_{m,n}}$ into ${\displaystyle {\mathfrak {M}}_{m',n'}}$ by identifying an element ${\displaystyle {\mathfrak {u}}\circ \ell _{n}}$ with the term

${\displaystyle \left({\mathfrak {u}}\circ \overbrace {e^{x}\circ \cdots \circ e^{x}} ^{n'-n}\right)\circ \ell _{n'}.}$

We then obtain ${\displaystyle \mathbb {T} ^{LE}}$ as the directed union

${\displaystyle \mathbb {T} ^{LE}=\bigcup _{m,n\in \mathbb {N} }\mathbb {T} _{m,n}^{LE}.}$

On ${\displaystyle \mathbb {T} ^{LE},}$ the right-composition ${\displaystyle \circ _{\ell }}$ with ${\displaystyle \ell }$ is naturally defined by

${\displaystyle \mathbb {T} _{m,n}^{LE}\ni \left(\sum f_{{\mathfrak {m}}\circ \ell _{n}}{\mathfrak {m}}\circ \ell _{n}\right)\circ \ell$ :=\sum f_{{\mathfrak {m}}\circ \ell _{n}}{\mathfrak {m}}\circ \ell _{n+1}\in \mathbb {T} _{m,n+1}^{LE}.}

Exponential and logarithm:

Exponentiation can be defined on ${\displaystyle \mathbb {T} ^{LE}}$ in a similar way as for log-free transseries, but here also ${\displaystyle \exp }$ has a reciprocal ${\displaystyle \log }$ on ${\displaystyle \mathbb {T} ^{LE,>}}$. Indeed, for a strictly positive series ${\displaystyle f\in \mathbb {T} _{m,n}^{LE,>}}$, write ${\displaystyle f={\mathfrak {m}}r(1+\varepsilon )}$ where ${\displaystyle {\mathfrak {m}}}$ is the dominant monomial of ${\displaystyle f}$ (largest element of its support), ${\displaystyle r}$ is the corresponding positive real coefficient, and ${\displaystyle \varepsilon$ :={\frac {f}{{\mathfrak {m}}r}}-1} is infinitesimal. The formal Hahn sum

${\displaystyle L(1+\varepsilon ):=\sum _{k\in \mathbb {N} }{\frac {(-\varepsilon )^{k}}{k+1}}}$

converges in ${\displaystyle \mathbb {T} _{m,n}^{LE}}$. Write ${\displaystyle {\mathfrak {m}}={\mathfrak {u}}\circ \ell _{n}}$ where ${\displaystyle {\mathfrak {u}}\in {\mathfrak {M}}_{m}}$ itself has the form ${\displaystyle {\mathfrak {u}}=x^{a}e^{\theta }}$ where ${\displaystyle \theta \in \mathbb {T} _{m,\succ }^{E}}$ and ${\displaystyle a\in \mathbb {R} }$. We define ${\displaystyle \ell ({\mathfrak {m}}):=a\ell _{n+1}+\theta \circ \ell _{n}}$. We finally set

${\displaystyle \log(f):=\ell ({\mathfrak {m}})+\log(c)+L(1+\varepsilon )\in \mathbb {T} _{m,n+1}^{LE}.}$

One may also define the field of log-exp transseries as a subfield of the ordered field ${\displaystyle \mathbf {No} }$ of surreal numbers.[4] The field ${\displaystyle \mathbf {No} }$ 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 ${\displaystyle F_{0}^{LE}=\mathbb {R} (\omega )}$, the subfield of ${\displaystyle \mathbf {No} }$ generated by ${\displaystyle \mathbb {R} }$ and the simplest positive infinite surreal number ${\displaystyle \omega }$ (which corresponds naturally to the ordinal ${\displaystyle \omega }$, and as a transseries to the series ${\displaystyle x}$). Then, for ${\displaystyle n\in \mathbb {N} }$, define ${\displaystyle F_{n+1}^{LE}}$ as the field generated by ${\displaystyle F_{n}^{LE}}$, exponentials of elements of ${\displaystyle F_{n}^{LE}}$ and logarithms of strictly positive elements of ${\displaystyle F_{n}^{LE}}$, as well as (Hahn) sums of summable families in ${\displaystyle F_{n}^{LE}}$. The union ${\textstyle F_{\omega }^{LE}=\bigcup _{n\in \mathbb {N} }F_{n}^{LE}}$ is naturally isomorphic to ${\displaystyle \mathbb {T} ^{LE}}$. In fact, there is a unique such isomorphism which sends ${\displaystyle \omega }$ to ${\displaystyle x}$ and commutes with exponentiation and sums of summable families in ${\displaystyle F_{\omega }^{LE}}$ lying in ${\displaystyle F_{\omega }}$.

• Continuing this process by transfinite induction on ${\displaystyle \mathbf {Ord} }$ beyond ${\displaystyle F_{\omega }^{LE}}$, taking unions at limit ordinals, one obtains a proper class-sized field ${\displaystyle \mathbb {R} \langle \langle \omega \rangle \rangle }$ canonically equipped with a derivation and a composition extending that of ${\displaystyle \mathbb {T} ^{LE}}$ (see Operations on transseries below).

• If instead of ${\displaystyle F_{0}^{LE}}$ one starts with the subfield ${\displaystyle F_{0}^{EL}:=\mathbb {R} (\omega ,\log \omega ,\log \log \omega ,\ldots )}$ generated by ${\displaystyle \mathbb {R} }$ and all finite iterates of ${\displaystyle \log }$ at ${\displaystyle \omega }$, and for ${\displaystyle n\in \mathbb {N} ,F_{n+1}^{EL}}$ is the subfield generated by ${\displaystyle F_{n}^{EL}}$, exponentials of elements of ${\displaystyle F_{n}^{EL}}$ and sums of summable families in ${\displaystyle F_{n}^{EL}}$, then one obtains an isomorphic copy the field ${\displaystyle \mathbb {T} ^{EL}}$ of exponential-logarithmic transseries, which is a proper extension of ${\displaystyle \mathbb {T} ^{LE}}$ equipped with a total exponential function.[7]

The Berarducci-Mantova derivation[8] on ${\displaystyle \mathbf {No} }$ coincides on ${\displaystyle \mathbb {T} ^{LE}}$ with its natural derivation, and is unique to satisfy compatibility relations with the exponential ordered field structure and generalized series field structure of ${\displaystyle \mathbb {T} ^{EL}}$ and ${\displaystyle \mathbb {R} \langle \langle \omega \rangle \rangle .}$

Contrary to ${\displaystyle \mathbb {T} ^{LE},}$ the derivation in ${\displaystyle \mathbb {T} ^{EL}}$ and ${\displaystyle \mathbb {R} \langle \langle \omega \rangle \rangle }$ is not surjective: for instance the series

${\displaystyle {\frac {1}{\omega \log \omega \log \log \omega \cdots }}:=\exp(-(\log \omega +\log \log \omega +\log \log \log \omega +\cdots ))\in \mathbb {T} ^{EL}}$

doesn't have an antiderivative in ${\displaystyle \mathbb {T} ^{EL}}$ or ${\displaystyle \mathbb {R} \langle \langle \omega \rangle \rangle }$ (this is linked to the fact that those fields contain no transexponential function).

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:

${\displaystyle \exp(x^{-1})=\sum _{n=0}^{\infty }{\frac {1}{n!}}x^{-n}\quad {\text{and}}\quad \log(x+\ell )=\ell +\sum _{n=0}^{\infty }{\frac {(x^{-1}\ell )^{n}}{n+1}}.}$

• Logarithm is defined for positive arguments.
• Log-exp transseries are real-closed.
• Integration: every log-exp transseries ${\displaystyle f}$ has a unique antiderivative with zero constant term ${\displaystyle F\in \mathbb {T} ^{LE}}$, ${\displaystyle F'=f}$ and ${\displaystyle F_{1}=0}$.
• Logarithmic antiderivative: for ${\displaystyle f\in \mathbb {T} ^{LE}}$, there is ${\displaystyle h\in \mathbb {T} ^{LE}}$ with ${\displaystyle f'=fh'}$.

Note 1. The last two properties mean that ${\displaystyle \mathbb {T} ^{LE}}$ is Liouville closed.

Note 2. Just like an elementary nontrigonometric function, each positive infinite transseries ${\displaystyle f}$ has integral exponentiality, even in this strong sense:

${\displaystyle \exists k,n\in \mathbb {N}$ :\quad \ell _{n-k}-1\leq \ell _{n}\circ f\leq \ell _{n-k}+1.}

The number ${\displaystyle k}$ is unique, it is called the exponentiality of ${\displaystyle f}$.

An original property of ${\displaystyle \mathbb {T} ^{LE}}$ is that it admits a composition ${\displaystyle \circ$ :\mathbb {T} ^{LE}\times \mathbb {T} ^{LE,>,\succ }\to \mathbb {T} ^{LE}} (where ${\displaystyle \mathbb {T} ^{LE,>,\succ }}$ is the set of positive infinite log-exp transseries) which enables us to see each log-exp transseries ${\displaystyle f}$ as a function on ${\displaystyle \mathbb {T} ^{LE,>,\succ }}$. Informally speaking, for ${\displaystyle g\in \mathbb {T} ^{LE,>,\succ }}$ and ${\displaystyle f\in \mathbb {T} ^{LE}}$, the series ${\displaystyle f\circ g}$ is obtained by replacing each occurrence of the variable ${\displaystyle x}$ in ${\displaystyle f}$ by ${\displaystyle g}$.

• Associativity: for ${\displaystyle f\in \mathbb {T} ^{LE}}$ and ${\displaystyle g,h\in \mathbb {T} ^{LE,>,\succ }}$, we have ${\displaystyle g\circ h\in \mathbb {T} ^{LE,>,\succ }}$ and ${\displaystyle f\circ (g\circ h)=(f\circ g)\circ h}$.
• Compatibility of right-compositions: For ${\displaystyle g\in \mathbb {T} ^{LE,>,\succ }}$, the function ${\displaystyle \circ _{g}:f\mapsto f\circ g}$ is a field automorphism of ${\displaystyle \mathbb {T} ^{LE}}$ which commutes with formal sums, sends ${\displaystyle x}$ onto ${\displaystyle g}$, ${\displaystyle e^{x}}$ onto ${\displaystyle \exp(g)}$ and ${\displaystyle \ell }$ onto ${\displaystyle \log(g)}$. We also have ${\displaystyle \circ _{x}=\operatorname {id} _{\mathbb {T} ^{LE}}}$.
• Unicity: the composition is unique to satisfy the two previous properties.
• Monotonicity: for ${\displaystyle f\in \mathbb {T} ^{LE}}$, the function ${\displaystyle g\mapsto f\circ g}$ is constant or strictly monotonous on ${\displaystyle \mathbb {T} ^{LE,>,\succ }}$. The monotony depends on the sign of ${\displaystyle f'}$.
• Chain rule: for ${\displaystyle f\in \mathbb {T} ^{LE}\times }$ and ${\displaystyle g\in \mathbb {T} ^{LE,>,\succ }}$, we have ${\displaystyle (f\circ g)'=g'f'\circ g}$.
• Functional inverse: for ${\displaystyle g\in \mathbb {T} ^{LE,>,\succ }}$, there is a unique series ${\displaystyle h\in \mathbb {T} ^{LE,>,\succ }}$ with ${\displaystyle g\circ h=h\circ g=x}$.
• Taylor expansions: each log-exp transseries ${\displaystyle f}$ has a Taylor expansion around every point in the sense that for every ${\displaystyle g\in \mathbb {T} ^{LE,>,\succ }}$ and for sufficiently small ${\displaystyle \varepsilon \in \mathbb {T} ^{LE}}$, we have

${\displaystyle f\circ (g+\varepsilon )=\sum _{k\in \mathbb {N} }{\frac {f^{(k)}\circ g}{k!}}\varepsilon ^{k}}$

where the sum is a formal Hahn sum of a summable family.

• Fractional iteration: for ${\displaystyle f\in \mathbb {T} ^{LE,>,\succ }}$ with exponentiality ${\displaystyle 0}$ and any real number ${\displaystyle a}$, the fractional iterate ${\displaystyle f^{a}}$ of ${\displaystyle f}$ is defined.[9]

The ${\displaystyle \left\langle +,\times ,\partial ,<,\prec \right\rangle }$ theory of ${\displaystyle \mathbb {T} ^{LE}}$ is decidable and can be axiomatized as follows (this is Theorem 2.2 of Aschenbrenner et al.):

• ${\displaystyle \mathbb {T} ^{LE}}$ is an ordered valued differential field.
• ${\displaystyle f>0\wedge f\succ 1\Longrightarrow f'>0}$
• ${\displaystyle f\prec 1\Longrightarrow f'\prec 1}$
• ${\displaystyle \forall f\exists g:\quad g'=f}$
• ${\displaystyle \forall f\exists h:\quad h'=fh}$
• Intermediate value property (IVP):

${\displaystyle P(f)<0\wedge P(g)>0\Longrightarrow \exists h:\quad P(h)=0,}$

where P is a differential polynomial, i.e. a polynomial in ${\displaystyle f,f',f'',\ldots ,f^{(k)}.}$

In this theory, exponentiation is essentially defined for functions (using differentiation) but not constants; in fact, every definable subset of ${\displaystyle \mathbb {R} ^{n}}$ is semialgebraic.

The ${\displaystyle \langle +,\times ,\exp ,<\rangle }$ theory of ${\displaystyle \mathbb {T} ^{LE}}$ is that of the exponential real ordered exponential field ${\displaystyle (\mathbb {R} ,+,\times ,\exp ,<)}$, which is model complete by Wilkie's theorem.

${\displaystyle \mathbb {T} _{\mathrm {as} }}$ 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 ${\displaystyle \mathbb {T} }$. (This conjecture is informal since we have not defined which isomorphisms of Hardy fields into differential subfields of ${\displaystyle \mathbb {T} }$ are permitted.) ${\displaystyle \mathbb {T} _{\mathrm {as} }}$ is conjectured to satisfy the above axioms of ${\displaystyle \mathbb {T} }$. 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 ${\displaystyle \mathbb {T} ^{LE}}$.[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]