2. Some preliminaries

Below are terminology, notations, and facts that will be used throughout.

As anticipated in the introduction, we shall use the following notation for the dominance relation: for $a,b$ in any ordered ring, we write

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$a\preceq b$ if $|a|\leq n|b|$ for some $n\in\mathbb{N}$ (a total partial order);
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$a\asymp b$ if $a\preceq b$ and $b\preceq a$ (an equivalence relation);
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$a\prec b$ if $a\preceq b$ and $b\not\preceq a$ ; equivalently, $n|a|<|b|$ for all $n\in\mathbb{N}$ (a strict partial order);
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$a\sim b$ if $a-b\prec a$ (an equivalence relation on the non-zero elements);
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$O(a)$ represents the convex class $\{b:b\preceq a\}$ , and similarly $o(a)$ represents $\{b:b\prec a\}$ ; both shall be used as in the big $O$ notation.

Remark 2.1.

Right composition by a fixed $x\in\mathbb{U}^{>\mathbb{R}}$ yields an ordered exponential field embedding. In particular, we also have for instance that $f\prec g$ holds if and only if $f\circ x\prec g\circ x$ , and as a special case, $f\prec T$ if and only if $f\circ x\prec x$ . The same holds for all of the above relations, since they are solely defined on the basis of the underlying ordered field structure. This will be used liberally in the proofs.

Fact 2.2.

As an ordered differential field, $\mathbb{R}\langle\!\langle T\rangle\!\rangle$ is an $H$ -field, namely $f>\mathbb{R}$ implies $f^{\prime}>0$ , otherwise $f=r+\varepsilon$ where $r^{\prime}=0$ (in fact, $r\in\mathbb{R}$ ) and $|\varepsilon|$ is smaller than all the constants (that is, $\varepsilon\prec 1$ ). This has numerous consequences, but the reader will only need to know that:

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if $1\not\asymp f$ , then $f\succeq g$ if and only if $f^{\prime}\succeq g^{\prime}$ , and if $f\succ g$ if and only if $f^{\prime}\succ g^{\prime}$ ;
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if $1\not\asymp f$ , then $f\asymp g$ if and only if $f^{\prime}\asymp g^{\prime}$ ;
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if $1\not\asymp f$ , then $f\sim g$ if and only if $f^{\prime}\sim g^{\prime}$ ;
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if $f\preceq 1$ , then $f^{\prime}\prec 1$ ;
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if $f\prec 1$ , $g\neq 0$ , $g\not\asymp 1$ , then $f^{\prime}\prec\frac{g^{\prime}}{g}$ .

Since $\mathbb{R}\langle\!\langle T\rangle\!\rangle$ is generated by $T$ , most arguments use induction on how elements are constructed starting from $T$ . We formalise this with the following rank. Since there is no risk of ambiguity, we shall abbreviate $\mathbb{J}_{\mathfrak{O}}=\mathbb{R}(\!(\mathfrak{O}^{>1})\!)_{\mathbf{On}}$ with just $\mathbb{J}$ .

Definition 2.3.

For any $f=\sum_{i<\alpha}r_{i}e^{\gamma_{i}}\in\mathbb{R}\langle\!\langle T\rangle\!\rangle$ , where each $r_{i}$ is a non-zero real number and $\gamma_{i}\in\mathbb{J}$ , we define the exponential rank $\mathrm{ER}(f)$ of $f$ to be the ordinal:

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$0$ if $f$ is a monomial of the form $\log^{\circ n}(T)$ for some $n\in\mathbb{N}$ , or if $f=0$ ;
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$\sup\{\mathrm{ER}(\gamma_{i})+1:i<\alpha\}$ otherwise.

This is clearly well defined (see [8] for more details).

Remark 2.4.

It is immediate from the definition that for $f,g\in\mathbb{R}\langle\!\langle T\rangle\!\rangle$ we have $\mathrm{ER}(f+g)\leq\max\{\mathrm{ER}(f),\mathrm{ER}(g)\}$ , unless $\mathrm{ER}(f)=\mathrm{ER}(g)=0$ , in which case $\mathrm{ER}(f+g)\leq 1$ . Similarly, $\mathrm{ER}(-f)\leq\mathrm{ER}(f)$ unless $\mathrm{ER}(f)=0$ , in which case $\mathrm{ER}(-f)=1$ . In particular $\mathrm{ER}(f-g)\leq\max\{\mathrm{ER}(f),\mathrm{ER}(g)\}$ unless $\mathrm{ER}(f)=\mathrm{ER}(g)=0$ .

We do not define Hahn fields here, but we refer the reader to any of the cited sources about transseries for details about the definition of sum, product, and order on them. We just remind the reader that the set of monomials appearing in a series $f=\sum_{i<\alpha}r_{i}\mathfrak{m}_{i}$ , meaning $\{\mathfrak{m}_{i}:i<\alpha\}$ , is called support of $f$ (note that by how we defined fields of transseries, the support of a single series is always a set even if the monomials range in a proper class). When $f\neq 0$ , we call the maximum $\mathfrak{m}_{0}$ of the support the leading monomial of $f$ , and we call $r_{0}\mathfrak{m}_{0}$ the leading term of $f$ . Note that by construction, $f\sim r_{0}\mathfrak{m}_{0}$ .

For clarity, we also remark again that $\exp$ and $\log$ have the following Taylor expansions for any $\varepsilon\prec 1$ in $\mathbb{U}$ :

\displaystyle\log(1+\varepsilon)

\displaystyle=\sum_{n=1}^{\infty}{(-1)}^{n+1}\frac{\varepsilon^{n}}{n},

\displaystyle\exp(\varepsilon)

\displaystyle=\sum_{n=1}^{\infty}\frac{\varepsilon^{n}}{n!}.

The infinite sum on the right is not a limit with respect to the topology induced by the order, but an algebraic operation in which each power $\varepsilon^{n}$ is expanded into a series, and then all series are summed term by term. For the details of how this is done, and why it is well defined, we defer again to the bibliography. Recall that $\mathbb{R}\langle\!\langle T\rangle\!\rangle$ is closed under infinite sums, and so in particular under the above ones.