Log-free 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.

Log-exp transseries

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

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

Direct construction of log-exp transseries

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 ${\displaystyle F_{\omega }}$.

Other fields of transseries

• 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).

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:

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

• 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 ${\displaystyle f}$.

Composition of transseries

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 ${\displaystyle g}$.

Theory of differential ordered valued differential field

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.

Theory of ordered exponential field

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.