1. Introduction

Transseries are a generalisation of formal power series that provide a framework for asymptotic analysis of certain classes of non-oscillating germs, one notable example being first return maps in Hilbert’s sixteenth problem. There are a few different flavours in the literature, such as LE-series and grid-based series (an incomplete list is [9, 12, 14, 15]), but for the most generality we will work with omega-series from [8], which are a little easier to define, and contain isomorphic copies of most other flavours.

Call field of transseries $\mathbb{T}$ a subfield of some (restricted) Hahn field

where the monomials $\mathfrak{M}$ form an ordered abelian group equipped with an isomorphism $\log:(\mathfrak{M},\cdot)\simeq\mathbb{J}_{\mathfrak{M}}\coloneqq(\mathbb{R}(\!(\mathfrak{M}^{>1})\!)_{\mathbf{On}},+)$ satisfying ${\log(\mathfrak{m})}^{n}<\mathfrak{m}$ for every $n\in\mathbb{N}$ (we call its inverse $\exp$ and abbreviate $e^{\gamma}=\exp(\gamma)$). We warn the reader that such $\mathfrak{M}$ must be a proper class; to avoid this issue, one can require, for instance, that the above ordinals $\alpha$ are countable. Plenty such fields exist, one notable example being Conway’s field of surreal numbers $\mathbf{No}$.

Now suppose that $\mathfrak{M}$ contains a log-atomic element $T\in\mathfrak{M}^{>1}$, namely such that $\log^{\circ n}(T)\in\mathfrak{M}$ for all $n\in\mathbb{N}$, where $\log^{\circ n}$ is the $n$-fold composition of $\log$. Then the field $\mathbb{R}\langle\!\langle T\rangle\!\rangle$ of omega-series in $T$ is the smallest subclass of $\mathbb{T}$ that contains $\mathbb{R}$, $T$ and is closed under taking $\exp$, $\log$, and sums like $\sum_{i<\alpha}r_{i}\mathfrak{m}_{i}$. $\mathbb{R}\langle\!\langle T\rangle\!\rangle$ is automatically of the form $\mathbb{R}(\!(\mathfrak{O})\!)_{\mathbf{On}}$ for some subgroup $\mathfrak{O}\leq\mathfrak{M}$. The choice of $T$ and of ambient $\mathbb{T}$ is irrelevant: all fields of omega-series are isomorphic to each other as fields of transseries.

By construction, $\mathbb{R}\langle\!\langle T\rangle\!\rangle$ contains, for instance, any formal Laurent series in $T^{-1}$, such as $T+1+T^{-1}+T^{-2}+\ldots$, but also series containing exponential terms, such as the asymptotic expansion of the $\Gamma$ function at $+\infty$:

By inspecting the constructions in the literature, one can easily verify that $\mathbb{R}\langle\!\langle T\rangle\!\rangle$ also contains (unique) copies of grid-based transseries ([12, 13]), of $\mathrm{LE}$-series ([9, 10]), and even $\mathrm{EL}$-series ([14]). One cannot forget mentioning that $\mathrm{LE}$-series and grid-based transseries have been subject of intense model theory investigations and, as differential ordered valued fields, they are model complete and have the same theory as any maximal Hardy field [1, 2].

Note that $\log$ can be extended naturally to a global function on the positive transseries: given $f\in\mathbb{T}^{>0}$, it can we written uniquely as $f=r\mathfrak{m}(1+\varepsilon)$, where $r\mathfrak{m}$ is the leading term of $f$, with $r\in\mathbb{R}^{>0}$ and $\mathfrak{m}\in\mathfrak{M}$, and $|\varepsilon|<\mathbb{R}^{>0}$, and we can define

where $\log(r)$ is standard real logarithm, and the infinite sum is always well defined, although we gloss over the details of what that means. Its inverse $\exp$ has a similar global extension given by a Taylor series. Since $\mathbb{R}\langle\!\langle T\rangle\!\rangle$ is closed under infinite sums, it is also closed under global $\exp$ and $\log$.

Just like traditional power series can be differentiated and composed, the same is true for omega-series. There is a unique derivation $f\mapsto f^{\prime}$ on $\mathbb{R}\langle\!\langle T\rangle\!\rangle$ ([15, 7]) satisfying $T^{\prime}=1$, $(e^{\gamma})^{\prime}=e^{\gamma}\gamma^{\prime}$ for any $\gamma\in\mathbb{J}$, and which is strongly $\mathbb{R}$-linear, namely

Likewise, given $g\in\mathbb{R}\langle\!\langle T\rangle\!\rangle$ with $g>\mathbb{R}$, there is a unique right composition map $f\mapsto f\circ g$ satisfying $T\circ g=g$, $e^{\gamma}\circ g=e^{\gamma\circ g}$ for $\gamma\in\mathbb{J}$, and which is strongly $\mathbb{R}$-linear

More generally, let $\mathbb{U}=\mathbb{R}(\!(\mathfrak{N})\!)_{\mathbf{On}}$ be a confluent field of transseries, namely such that for every $x\in\mathbb{U}^{>\mathbb{R}}$ there is $n\in\mathbb{N}$ such that the leading term of $\log^{\circ n}(x)$ is log-atomic (this holds in $\mathbb{R}\langle\!\langle T\rangle\!\rangle$, as by construction, $\log^{\circ n}(x)$ has leading monomial of the form $\log^{\circ k}(T)$ for $n$ sufficiently large). Note that $\mathbb{U}$ may be a new field, different from $\mathbb{T}$, although it will also contain copies of $\mathbb{R}\langle\!\langle T\rangle\!\rangle$ obtained by replacing $T$ with log-atomic monomials of $\mathbb{U}$. Then for every $x\in\mathbb{U}^{>\mathbb{R}}$ there is a unique map $f\mapsto f\circ x$ satisfying the above properties with $x$ in place of $g$ (while the uniqueness is obvious, the existence is technically challenging, see [15, 8]; while [8] is stated for $\mathbf{No}$, and seemingly using the assumption ‘T4’, the proof therein only uses confluence). Surreal numbers are an example of confluent field ([7]).

Finally, we remark that composition and derivation are compatible with global $\exp$, namely $\exp(f)^{\prime}=\exp(f)f^{\prime}$ and $\exp(f)\circ g=\exp(f\circ g)$ for any $f\in\mathbb{R}\langle\!\langle T\rangle\!\rangle$, and also with each other, in the sense of the chain rule: for any $f,g\in\mathbb{R}\langle\!\langle T\rangle\!\rangle$ with $g>\mathbb{R}$ we have $(f\circ g)^{\prime}=(f^{\prime}\circ g)g^{\prime}$. Moreover, composition is compatible ‘with itself’ in the sense that it is associative: $(f\circ g)\circ x=f\circ(g\circ x)$ for any $f\in\mathbb{R}\langle\!\langle T\rangle\!\rangle$, $g\in\mathbb{R}\langle\!\langle T\rangle\!\rangle^{>\mathbb{R}}$, $x\in\mathbb{U}^{>\mathbb{R}}$.

It is immediate from the definition that right composition by some $x\in\mathbb{U}^{>\mathbb{R}}$ must be an ordered field embedding: by construction, the map $f\mapsto f\circ x$ preserves (infinite) sums, scalar multiplication by $\mathbb{R}$, $\exp$, $\log$, and it quickly follows that it preserves multiplication and ordering.

However, the properties of left composition by some $f\in\mathbb{R}\langle\!\langle T\rangle\!\rangle$, namely of the map $x\mapsto f\circ x$, have not been explored as much. Since transseries have been introduced to provide asymptotic expansions of real functions, we expect these maps to share at least the most basic properties. For instance, we expect the map to be strictly increasing exactly when $f^{\prime}>0$. To the best of my knowledge, this only appears in a rarely cited preprint of Edgar [11, Prop. 4.9], for $\mathrm{LE}$-series only, under the heading ‘Simpler proof needed’, and with an ‘overly-involved proof’.

Addressing Edgar’s need, this note gives a short, self-contained proof of monotonicity for general omega-series. We also spell out a small set of inductive assumptions, so that the method can be reapplied more easily to fields larger than $\mathbb{R}\langle\!\langle T\rangle\!\rangle$, in particular with an eye to the hyperseries of [3], which generalise transseries by adding transexponential functions. From now on, let $\mathbb{U}$ be a fixed confluent field of transseries, for instance $\mathbb{U}=\mathbf{No}$, or more generally a field of transseries equipped with a left composition by omega-series that preserves infinite sums, scalar multiplication by $\mathbb{R}$, $\exp$, and $\log$.

Since $\mathbb{R}\langle\!\langle T\rangle\!\rangle$ is generated from $T$, a naive proof by induction would be as follows: the conclusion is true for the base case $f=T$, $f=\log(T)$, …; if $f=\sum_{i<\alpha}r_{i}e^{\gamma_{i}}$, we are free to assume the conclusion for each map $x\mapsto\gamma_{i}\circ x$, and we would like to deduce that $x\mapsto f\circ x$ is also monotonic. This works very well for $f\in\mathbb{J}_{\mathfrak{O}}^{>0}$, and we do so in Proposition 3.1, but the inductive assumption falls short for the general case. We also need the following very weak form of mean value property: given $\gamma\in\mathbb{J}_{\mathfrak{O}}^{>0}$ and $x,y\in\mathbb{U}$ with $x>y>\mathbb{R}$, we have

where $a\preceq b$ means $|a|\leq n|b|$ for some $n\in\mathbb{N}$ and $a\prec b$ means $n|a|<|b|$ for all $n\in\mathbb{N}$. The condition is almost trivial to verify in the base case of $T$, $\log(T)$, $\ldots$, as the associated functions $x\mapsto\log^{\circ n}(x)$ are all strictly increasing and concave. The rest of the induction is then relatively straightforward (see Lemma 4.1), and it leads to a fairly short proof of Theorem A.

Since omega-series admit compositional inverses by [6], namely for every $f\in\mathbb{R}\langle\!\langle T\rangle\!\rangle^{>\mathbb{R}}$ there is a (unique) $f^{\operatorname{inv}}\in\mathbb{R}\langle\!\langle T\rangle\!\rangle^{>\mathbb{R}}$ such that $f\circ f^{\operatorname{inv}}=f^{\operatorname{inv}}\circ f=T$, we can immediately deduce the following.

$\mathrm{LE}$-series are closed under composition and admit compositional inverses within $\mathrm{LE}$ ([10]), thus the intermediate value property applies even for $x\mapsto f\circ x$ seen as a function from $\mathrm{LE}^{>\mathbb{R}}$ to $\mathrm{LE}$.

Given Theorem A, it also becomes relatively easy to prove the following form of Taylor’s theorem. We write $a\asymp b$ to mean $a\preceq b$ and $b\preceq a$, and $O(b)$ to represent an unspecified element of the set $\{a:a\preceq b\}$.

The restriction $f\not\asymp T^{k}$ is a mere technicality: since the derivatives of $T^{k}$ vanish after $k+1$ steps, one must look at $f^{(k+1)}$ to determine the radius of validity of the approximation, which results in a slightly more complicated but functionally equivalent statement (see Corollary 5.6).

Other versions of the Taylor theorem exist in the literature, but they focus on the equality $f\circ(x+\delta)=\sum_{i=0}^{\infty}\frac{f^{(i)}\circ x}{i!}\delta^{i}$, which in general holds for a much smaller class of $\delta$’s (see for example [10, §6], [8, Prop. 7.13]). What is notable here is that the Taylor approximation is valid for the essentially largest possible meaningful class of $\delta$’s. Indeed, when $\delta$ is small as required by Theorem C, the sequence $(f^{(n)}\circ x)\delta^{n}$ is weakly decreasing for $\prec$, and even strictly decreasing if we require $\delta\prec x$ and $(f^{\dagger}\circ x)\delta\prec 1$, whereas it is strictly increasing if $\delta\succ x$ or $(f^{\dagger}\circ x)\delta\succ x$, making the error terms larger rather than smaller (see Corollary 5.3). The conclusion may hold or fail in the remaining case $x+\delta\prec x$ depending on $f$ (Remark 5.5). In separate work with Bagayoko [5], we tackle the equality $f\circ(x+\delta)=\sum_{i=0}^{\infty}\frac{f^{(i)}\circ x}{i!}\delta^{i}$ for omega-series, and show it holds under $\delta\prec x$ and the strict inequality $(\mathfrak{m}^{\dagger}\circ x)\delta\prec 1$ with respect to every monomial $\mathfrak{m}$ appearing in $f$, rather than just $f$, expanding and generalising the numerous existing variants of the same result.

Several questions remain open. Notably, while we have seen that omega-series and $\mathrm{LE}$-series, when interpreted as functions, have the intermediate value property, we are not claiming that they have the mean value property: given $z$ such that $f^{\prime}\circ z=\frac{f\circ x-f\circ y}{x-y}$, Theorem A does not seem to imply directly that $z$ is between $x$ and $y$. This is related to whether $f^{\prime\prime}\geq 0$ implies that the function $x\mapsto f\circ x$ is convex, namely that $f^{\prime}\circ y\leq\frac{f\circ x-f\circ y}{x-y}$ for $x>y$, and it may require a new argument.

Going beyond omega-series, it would be interesting to know if composition of elements in hyperserial fields by hyperseries ([3]) is also monotonic as in Theorem A, and deduce the corresponding generalisation of Theorem C.

In this context, the first question is whether the weak mean value property used in the proof holds for the basic hyperlogarithmic monomials ‘$\ell_{\alpha}$’, which represent, in a suitable sense, the ‘$\alpha$-th iterates of $\log$’. While the condition is satisfied locally, by the local Taylor expansion of the transfinite iterates of $\log$, it does not seem obvious at a first glance that it has to hold globally. For the usual $\log$, there is no issue because of the functional equation $\log(xy)=\log(x)+\log(y)$, but from $\ell_{\omega}$ onwards functions are less well behaved. In principle, this may require additional restrictions on the growth properties in the definition of hyperserial fields, and one should check if they are already satisfied in the surreal model of [4].

A similar, and potentially equivalent consideration emerges from Proposition 3.1 (monotonicity for $\gamma\in\mathbb{J}^{>0}$): its proof relies on the fact that for $x>y>\mathbb{R}$ and $z>w>\mathbb{R}$, if $y>w$ and $x-y>z-w$, then $\frac{\exp(x)}{\exp(y)}>\frac{\exp(z)}{\exp(w)}>1$. Once again, this property seems easy to verify locally for transfinite iterations of $\exp$, but it is less clear whether it can be extended globally, due to the lack of a functional equation like $\exp(x+y)=\exp(x)\exp(y)$.