Table of Contents

1. Introduction

In history, formal power series are used extensively for finding the resolution of differential equations. If the resulting power series is convergent, it gives rise to a germ which can be analytically continued to (multi-valued) functions on a Riemann surface. However, very often, the power series we found from solving a differential equation is divergent, then it is not clear a prior how to attach reasonable sums to them.

The modern theory of resummation was developed systematically by Stieltjes, Borel and Hardy, who invented some resummation methods which are stable under the common operators of analysis. Later, Poincare established the equivalence between computations with formal power series and asymptotic expansions. Newton, Borel and Hardy were all aware of the systematic aspects of their theories and they consciously tried to complete their framework so as to capture as much of analysis as possible. The great unifying theory nevertheless had to wait until the late 20-th century and Ecalle’s work on transseries and Dulac’s conjecture.

Transseries have found significant applications in various areas of physics, particularly in high-energy physics. They are employed as algebraic tools to investigate self-consistent Dyson–Schwinger equations, which are integral equations that arise in the field of quantum field theory, specifically in Yukawa theory and quantum electrodynamics 1. These equations are pivotal in understanding the interactions of particles and fields at a fundamental level.

In the realm of general relativity, transseries are applied to asymptotic analysis. General relativity stands as one of the cornerstones of modern physics, governing the laws of gravitation and the dynamics of large-scale structures in the universe . By applying transseries in this domain, researchers can gain insights into the asymptotic behavior of gravitational fields and the dynamics of spacetime.

Furthermore, transseries are used in the extraction of non-perturbative physics from perturbation theory through resurgence and alien calculus. Perturbation theory is a fundamental tool in quantum mechanics and quantum field theory, allowing for the approximation of complex systems. The non-perturbative effects are those that cannot be captured by perturbation theory alone, and transseries help to identify and understand these effects 2.

Additionally, in the context of integrable, asymptotically free field theories, transseries have applications in studying the free energy of such systems when coupled to a conserved charge. These studies are significant in high-energy physics, particularly in understanding the thermodynamics and statistical mechanics of particle systems 3.

These examples showcase the versatility and importance of transseries in advancing the understanding of fundamental physics, from the microscale of particle interactions to the macroscale of cosmic phenomena.

Define a ordered group ${\frak G}$ (frak G) of transmonomials. Define a differential field $\mathbb{T}$ of transseries. Transmonomials are generalizations of monomials in polynomials, by including exponential and logarithmic. In this note and that follows, we will consider the limit where $x\to \infty$.

Let’s start with exponents first.

Log-free transmonomials. They are of form

\[x^{b}e^{ L },\quad b\in \mathbb{R},\; L \in \text{large log-free transseries.}\]

For example, the following are all log-free transmonomials,

\[x^{-1},\; x^{\pi}x^{x^{\sqrt{ 2 }}-3x},\; e^{ \sum_ {i}x^{-1}e^{ x } },etc.\]

The multiplication is defined in the obvious way. The group identity is just $1$.

We define a binary relation $\gg$, read “far larger than”. Keep in mind that we assumed $x\to \infty$. So how does this “far larger than” work? We compare the exponents $L$ in $e^{ L }$ first, if the exponent is large then $e$ to the exponent is far larger; if they have same exponents, then we compare the power of $x$, namely $x^{b}$, whichever with larger $b$ is far larger then others. To be specific,

\[x^{b_ {1}}e^{ L_ {1} } \gg x^{b_ {2}}e^{ L_ {2} }\quad \text{ if } L_ {1} > L_ {2} \;\lor\; (L_ {1}=L_ {2}\;\land\; b_ {1}>b_ {2} ),\]

where $\lor$ is logic or. For example, $x^{-5}\gg x^{20}e^{ -x }$ since $x^{-5}=x^{-5}e^{ 0 }$ and $0>-x$.

Log-free transseries. A log-free transseries $T$ is a formal sum of log-free monomials ${\frak g}$,

\[T = \sum_ {i} c_ {i} {\frak g}_ {i},\quad c_ {i} \in \mathbb{R} .\]

We require the order of transmonomials be such that, each ${\frak g_ {i}}$ is far smaller than all previous terms, namely they appear in descending orders. This is similar to the case of regular polynomials where we usually put the highest powers at first.

The transseries $T$ is said to be purely large if all transmonomials ${\frak g}_ {i}$ are far larger than $1$ (not $0$), namely ${\frak g_ {i}}\gg 1 \;\forall i$. $T$ is said to be small if all ${\frak g}_ {i}\ll 1$ (why isn’t it called purely small?). The largest (in the sense of far larger than) transmonomial is called the dominant term, let’s call it $c_ {0}{\frak g}_ {0}$. If the dominant term has positive coefficients, $c_ {0}>0$, then $T$ is said to be positive. This enables us to compare the size of two transseries $S,T$, we say $S>T$ if $S-T>0$. So we just need to compare their dominant terms.

We consider only transmonomials and transseries of “finite exponential height”. For example, we don’t want

\[e^{ x^{x^{x\dots}} }.\]

The differentiation of $T$ with respect to $x$ is defined the usual way.

Next let’s include logarithmic. $\log$ acting $m$ times is denoted $\log_ {m}x$ or $\log_ {(m)}x$, namely

\[\log_ {(m)}x = \log \dots \log x,\quad m\; \log .\]

A general transseries is obtained by substitution of some $\log_ {m}x$ for $x$ in a log-free transseries.

Every nonzero transseries has a multiplicative inverse. This is similar to formal power series.

A lot of functions can now be regarded as a transseries. For example, e hyperbolic sine is a two-term transseries.

2. Formal Constructions

In mathematics, the move towards higher levels of formality entails adopting rigorous and precise language, definitions, and proofs, which brings clarity and precision, ensuring that mathematical concepts are universally understood and applied correctly. It allows for the development of solid, gap-free proofs, having a deeper understanding of mathematical structures and providing a robust foundation for complex theories. This precision in communication is critical in a global context, where scientists from different fields rely on universally recognized formalisms to understand each other effectively.

However, this precision comes at a cost. It can make the subject less accessible to beginners (like myself), potentially hindering educational and interdisciplinary work. A focus on stringent formalism might even inhibit creative thinking, as the rigidity of formal proofs could constrain the exploratory, intuitive processes that often drive mathematical discovery. Moreover, the lengthy and detailed nature of formal proofs can make mathematical work less efficient, both in terms of personal understanding and communication with others. There’s also the risk of diminishing intuition, which is a crucial aspect of mathematical thought, particularly in the preliminary stages of research. It is definitely crucial to have a balance between concrete examples and general formalism, what is I hope to achieve in this note.

The set of monomials ${\frak G}$ form a field, which is also a group if we focus on multiplication alone. ${\frak G}$ is not finitely generated, to see this consider the finitely generated group with generator

\[\mu_ {1},\mu_ {2},\dots,\mu_ {n},\]

the generated group has elements of form

\[\left\lbrace \mu_ {1}^{k_ {1}}\times \mu_ {2}^{k_ {2}}\times \dots \times \mu_ {n}^{k_ {n}} \,\middle\vert\, k_ {1},\dots,k_ {n} \in \mathbb{Z} \right\rbrace .\]

Note that the exponents must be integers.

Let’s use capital letters to denote a set of indices, for example define

\[K := (k_ {1},k_ {2},\dots,k_ {n})\]

then

\[\mu_ {1}^{k_ {1}}\dots \mu_ {n}^{k_ {n}} =: \mu^{K}.\]

This will save some writing. The problem is that $\mu^{K}$ can also be interpreted as $\mu^{k_ {1}k_ {1}\dots k_ {n}}$, but it should be clear from the context.

We will assume that all the generator $\mu \ll 1$. We will think of these as “ratios” between one term of a series and the next. A ratio set is a finite set of small monomials.

Let $k \in \mathbb{Z}^{n}$ be an $n$-tuple of integers, it form a group under addition. Let $p$ be another such $n$-tuple, one say

\[k \leq p \quad \text{ iff } k_ {i} \leq p _ {i} \;\forall\; i,\]

where $k_ {i}$ is the $i$-th component of $k$.

$J_ {m}$ is a partially ordered set. To be specific, a partially ordered set (or poset) is a set equipped with a binary relation that captures a certain level of order or precedence among its elements. This binary relation is denoted by $\leq$ and must satisfy the following properties for any elements $a, b$, and $c$ in the set:

  1. Reflexivity: For all elements a in the set, $a\leq a$. In other words, every element is related to itself;
  2. Antisymmetry: If $a \leq b$ and $b \leq a$, then $a = b$. This property ensures that no two distinct elements are related in both directions;
  3. Transitivity: If $a \leq b$ and $b \leq c$, then $a \leq c$. This property means that if there’s an order relationship between $a$ and $b$, and another between $b$ and $c$, there’s also an order relationship between $a$ and $c$.

A partially ordered set does not require every pair of elements to be comparable; that is, it’s possible for $a$ and $b$ to be in the set without $a\leq b$ or $b \leq a$ being true. This distinguishes partially ordered sets from totally ordered sets, where every pair of elements is comparable.

For $m\in \mathbb{Z}^{n}$, define

\[J_ {m} := \left\lbrace k \in \mathbb{Z}^{n} \,\middle\vert\, k\geq m \right\rbrace .\]

Apparently $m \in J_ {m}$. The sets $J_ {m}$ will be used to define grids of monomials. For example, if

\[\mu_ {1} = \frac{1}{x},\quad \mu_ {2}=e^{ -x }\]

comprise the ratio group (recall that each element of a ratio group is required to be small), then we can define a grid (about which we will say more later)

\[\left\lbrace \mu^{k} \,\middle\vert\, k\in J_ {(-1,2)} \right\rbrace\]

which is the same as

\[\left\lbrace \mu^{k}=\mu_ {1}^{k_ {1}} \cdot \mu_ {2}^{k_ {2}} \,\middle\vert\, (k_ {1},k_ {2})\geq (-1,2) \right\rbrace .\]

In our convention, the set of natural numbers $\mathbb{N}$ include zero.

2.1 Basics Order Theory

The basic unit of analysis in order theory is binary relation. Given a set $S$ with elements $s$’s, and a binary relation denoted $\mathscr{R}$, any two elements $s_ {1}$ and $s_ {2}$ either have this relation or they don not. If they the specified relation, we write $s_ {1}\mathscr{R}s_ {2}$. Note that the order also matters, $s_ {1}\mathscr{R}s_ {2}$ is in general not the same as $s_ {2}\mathscr{R}s_ {1}$, just think of $\leq$ as an example.

The asymptotic magnitude of transseries refers to the growth rate of a transseries as its argument tends to infinity, for example, how fast does $e^{ x }$ grows as $x\to \infty$. The asymptotic magnitude can be regarded as a kind or ordering.

We have already introduced ordering and partial ordering. An easy way to memorize ordering is to regard ordering as categories. Take $a\leq b$ for example, we can regard $a$ and $b$ as objects in a category, and regard $a\leq b$ as $a\to b$, an arrow from $a$ to $b$. For partial ordering, not all pairs need to me comparable, it means that if regarded as a category, not all pairs of objects need to have arrows. In category theory, every object is required to have an identity map, $a\to a$, this corresponds to the reflexivity of ordering, $a\leq a$ always. Antisymmetry means that $a\to b$ and $b\to a$ implies $a=a$. Transitivity for ordering is automatically taken care of in the language of category theory, by composition of arrow.

Now let’s introduce quasi ordering. A quasi-order (or preorder) is a binary relation that is reflexive and transitive. It is a generalization of a partial order, but it does not necessarily satisfy antisymmetry. A quasi-order is nothing but a category, where arrows represent the ordering relation. Here are the formal properties:

  1. Reflexive: For every element $a$ in a set $S$, $a \leq a$.

  2. Transitive: For every $a, b, c$ in $S$, if $a \leq b$ and $b \leq c$, then $a \leq c$.

As an example, consider the set $S = \lbrace a, b, c\rbrace$ with the relation $\leq$ defined as follows:

  • $a \leq a, b \leq b, c \leq c$ (reflexivity)
  • $a \leq b$
  • $b \leq c$
  • $a \leq c$ (transitivity)

This relation is a quasi-order because it is reflexive and transitive. However, it is not necessarily antisymmetric. For instance, if we had both $a \leq b$ and $b \leq a$, but $a \neq b$, this would still be a quasi-order but not a partial order.

  • Partial Order: A binary relation that is reflexive, transitive, and antisymmetric.
  • Quasi-Order: A binary relation that is reflexive and transitive, but not necessarily antisymmetric.

A common real-life example of a quasi-order is the divisibility relation among integers. For a set of integers $\lbrace1, 2, 3, 4\rbrace$:

  • $1 \mid 1, 2 \mid 2, 3 \mid 3, 4 \mid 4$ (reflexivity)
  • $1 \mid 2, 1 \mid 3, 1 \mid 4$
  • $2 \mid 4$
  • $1 \mid 4$ (transitivity)

The divisibility relation is reflexive and transitive, but not antisymmetric because, for example, $1 \mid 2$ and $2 \mid 1$ are not both true unless the integers are the same.

As an example for quasi-order that is not necessarily a partial order is the relation of “being at least as easy to learn as” among different subjects. This relation can be reflexive and transitive, but it might not be antisymmetric because two subjects can be equally easy to learn without being identical subjects.

If a partial order also satisfies the condition of completeness, meaning that any pair of the set is comparable, then it is called a total order, or a linear order.

A set is said to be well-ordered if it is equipped with a total order such that every non-empty subset has a least element. To be exact, a set $S$ with a total order $\leq$ is well-ordered if every non-empty subset $T \subseteq S$ has a least element. That is, there exists an element $m \in T$ such that for all $t \in T$, $m \leq t$.

A classic example of a well-ordered set is the set of natural numbers $\mathbb{N}$ with the usual order $\leq$. Given any non-empty subset of $\mathbb{N}$, there is always a smallest element.

The well-ordering theorem states that every set can be well-ordered. This theorem is equivalent to the Axiom of Choice and Zorn’s Lemma in the sense that any one of these statements implies the others in the context of Zermelo-Fraenkel set theory.

If we regard ordered sets as objects in an category, no matter quasi or partial or well ordered; a morphism between these objects should preserve the ordering. A morphism is also called an arrow. Morphisms can be used to compare different orderings, as we will see below.

Let $\leq_1$ and $\leq_2$ be two binary relations on a set $S$. The relation $\leq_1$ is said to be finer than $\leq_2$ if for all $a, b \in S$,

\[a \leq_1 b \implies a \leq_2 b.\]

Namely there exists a morphism $\phi:(S,\leq_ {1})\to(S,\leq_ {2})$.

Ex. Let $S = \lbrace1, 2, 3\rbrace$, and consider two different orderings on the power set of $S$ (the set of all subsets of $S$), 1) the subset inclusion Ordering $\subseteq$ and 2) the lexicographic ordering $leq_{\text{lex}}$. For example, for the power set $\mathcal{P}(S) = \lbrace\emptyset, \lbrace1\rbrace, \lbrace2\rbrace, \lbrace3\rbrace, \lbrace1, 2\rbrace, \lbrace1, 3\rbrace, \lbrace2, 3\rbrace, \lbrace1, 2, 3\rbrace\rbrace$, the inclusion ordering is defined as

  • $\emptyset \subseteq \lbrace1\rbrace$
  • $\lbrace1\rbrace \subseteq \lbrace1, 2\rbrace$
  • $\lbrace2\rbrace \subseteq \lbrace2, 3\rbrace$
  • etc.

The lexicographic ordering, on the other hand, is based on comparing the elements of subsets lexicographically, treating subsets as ordered tuples of their elements, and we compare first the first element, if they are equal we compare the second, etc. For example, we can define an ordering such that:

  • $\emptyset \leq_{\text{lex}} \lbrace1\rbrace$
  • $\lbrace1\rbrace \leq_{\text{lex}} \lbrace1, 2\rbrace$
  • $\lbrace1\rbrace \leq_{\text{lex}} \lbrace1, 3\rbrace$
  • $\lbrace1, 2\rbrace \leq_{\text{lex}} \lbrace1, 3\rbrace$
  • $\lbrace2\rbrace \leq_{\text{lex}} \lbrace2, 3\rbrace$
  • etc.

The subset inclusion ordering is finer than the lexicographic ordering on the power set $\mathcal{P}(S)$. This means that whenever $A \subseteq B$, it must also be true that $A \leq_{\text{lex}} B$, but the converse does not necessarily hold. Comparing $\lbrace1\rbrace$ and $\lbrace1, 2\rbrace$,

  • $\lbrace1\rbrace \subseteq \lbrace1, 2\rbrace$
  • $\lbrace1\rbrace \leq_{\text{lex}} \lbrace1, 2\rbrace$

Comparing $\lbrace1, 2\rbrace$ and $\lbrace1, 3\rbrace$:

  • Neither $\lbrace1, 2\rbrace \subseteq \lbrace1, 3\rbrace$ nor $\lbrace1, 3\rbrace \subseteq \lbrace1, 2\rbrace$
  • But in lexicographic ordering, $\lbrace1, 2\rbrace \leq_{\text{lex}} \lbrace1, 3\rbrace$.

3. Dickson’s lemma

It turns out that the set $J_ {m}$ is well-partially-ordered, sometimes called Noetherian. A partially ordered set (poset) is said to be well-partially-ordered if it satisfies two conditions:

  • It contains no infinite strictly descending sequences. This means there cannot be an infinite sequence of elements $a_1, a_2, a_3, \ldots$ in the set such that $a_1 > a_2 > a_3 > \ldots$.
  • It contains no infinite antichains. An antichain is a subset of the poset in which no two distinct elements are comparable. In a well-partially-ordered set, there cannot be an infinite set of elements where none are comparable to each other.

On the other hand, a poset is called Noetherian if it satisfies the descending chain condition (DCC), which states that every descending sequence of elements eventually stabilizes. In other words, there cannot be an infinite strictly descending sequence of elements in the set.

The primary difference between the two concepts is that being well-partially-ordered is a stronger condition than being Noetherian. While both require the absence of infinite strictly descending sequences (the Noetherian property), being well-partially-ordered also requires the absence of infinite antichains. Therefore, every well-partially-ordered set is Noetherian, but not every Noetherian set is well-partially-ordered.

Proposition. If $E \subseteq J_ {m}$ and $E \neq \emptyset$, then there is a minimal element $k_ {0} \in E$.

Proposition. Let $E$ from the previous proposition be infinite, then there is an infinite sequence $k_ {i}\subset E$ such that $k_ {0}<k_ {1}<\dots$.

Proposition. For the same $E$, the set of all the minimal elements $\text{min}(E)$ is finite. For every element $k\in E$ there is a $k_ {0} \in \text{min}(E)$ such that $k_ {0} \le k$.

4. Convergence of sets

First let’s introduce the symmetric difference of two sets, which is a mathematical operation that results in a new set containing elements that are in either of the two sets, but not in their intersection. In other words, it combines the elements of each set that are not shared by both. The symmetric difference is denoted by the symbol $\Delta$.

Formally, if you have two sets $A$ and $B$, their symmetric difference $A \Delta B$ is defined as:

\[A \Delta B = (A - B) \cup (B - A).\]

Here, $A - B$ represents the set of elements in $A$ but not in $B$, and $B - A$ represents the set of elements in $B$ but not in $A$. The union of these two sets ($\cup$) forms the symmetric difference.

Another way to express this is using the union and intersection of sets:

\[A \Delta B = (A \cup B) - (A \cap B).\]

By subtracting the intersection from the union, you’re left with only those elements that are exclusively in either $A$ or $B$, but not in both.

The symmetric difference has several interesting properties:

  • It is commutative: $A \Delta B = B \Delta A$.
  • It is associative: $(A \Delta B) \Delta C = A \Delta (B \Delta C)$.
  • The symmetric difference of a set with itself is the empty set: $A \Delta A = \emptyset$.
  • The symmetric difference of a set with the empty set is the set itself: $A \Delta \emptyset = A$.

Now let’s define the convergence of a sequence of sets $E_ {i}, i\in I$, where $I$ is an infinite index set. For each $i\in I$, let $E_ {i} \in \mathbb{Z}^{n}$. We say the family $E_ {i,\,i\in I}$ is point-finite if each $p\in \mathbb{Z}^{n}$ belong to $E_ {i}$ for only finite many $i$ (could be zero).

Let $m\in\mathbb{Z}^{n}$ and define $J_ {m}$ as before. If the “limit” of $E_ {i}$ is empty, we write

\[E_ {i} \xrightarrow{m}\emptyset,\quad \text{iff } E_ {i} \subset J_ {m} \text{ and } (E_ {i}) \text{ is point-finite.}\]

More generally, we write

\[E_ {i} \to \emptyset\]

if there exist $m\in\mathbb{Z}^{n}$ such that $E_ {i}\xrightarrow{m}\emptyset$.

Now we can generalize the situation to other than the empty set. To do that we need the concept of symmetric difference. Intuitively, if the limit of $E_ {i}$ is another set $E$, then there should be infinite sets $E_ {i, \, i>j}$ for some $j$ such that they all “tend to” contain $E$. It implies that $E_ {i}-E$ should tend to be empty, for if there is some element $e$ in the limit of $E_ {i}-E$ then $e$ should be in the limit, too. On the other hand, the limit $E$ shouldn’t contain any more element that is not in $E_ {i}$’s. For example, if for all $i>j$, some element $e’$ is not in $E_ {i,i>j}$ then it should not be contained in the limit $E$ as well. These very hand-waving arguments inspires us to define the limit of $E_ {i}$ as follows.

We write

\[E_ {i}\xrightarrow{m} E,\quad \text{iff }E_ {i} \subset J_ {m} \text{ and } E_ {i}\Delta E\xrightarrow{m}\emptyset .\]

All the properties we want for a limit of $E_ {i}$ are concisely contained in the condition that $E_ {i}\Delta E$ tend to the empty set. Also, we simply write

\[E_ {i}\to E\]

if there exists some $m$ such that $E_ {i}\xrightarrow{m}E$ and we don’t care about the details of $m$.

Equivalently, we could say that $(E_ {i}\Delta E)$ is point finite.

Example. Consider $\mathbb{Z}^{n}=\mathbb{Z}^{1}=\mathbb{Z}$. Let $E_ {i} =\left\lbrace i,i+1,\dots \right\rbrace$ for $i\in \mathbb{N}$. Then the sequence $E_ {i}$ is point-finite. We have

\[E_ {i}\to \emptyset .\]

But if we let $F_ {i}=\left\lbrace -i \right\rbrace$, then even though $F_ {i}$ is point-finite but $F$ can not be contained in some $J_ {m}$ so it does not converge.

Notation. Let $k = (k_ {1},k_ {2},\dots,k_ {n})$ then define $\left\lvert k \right\rvert:=k_ {1}+k_ {2}+\dots+k_ {n}$.

For two pints $p,q$ in $J_ {m}$, we can introduce the concept of distance by defining

\[d(p,q):= 2^{-\left\lvert p-q \right\rvert }\]

This reminds me of Krull topology.

For two sets $E,F\subset J_ {m}$, define

\[d(E,F):= \sum_ {k\in E\Delta F} 2^{-\left\lvert k \right\rvert },\]

then for any $E_ {i}\subset J_ {m}$, we have $E_ {i}\to E$ iff $d(E_ {i},E)\to 0$. And $d$ is a metric on the grid $J_ {m}$.