# The fundamental theorem of tropical differential algebra for formal Puiseux
series

Sebastian Falkensteiner<EMAIL_ADDRESS>Max Planck
Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig,
Germany  and  Stefano Mereta<EMAIL_ADDRESS>
###### Abstract.

The fundamental theorem of tropical differential algebra has been established
for formal power series solutions of systems of algebraic differential
equations. It has been shown that the direct extension to formal Puiseux
series solutions fails. In this paper, we overcome this issue by transforming
the given differential system and such a generalization is presented.
Moreover, we explain why such transformations do not work for generalizing the
fundamental theorem to transseries solutions, but show that one inclusion
still holds for this case even without using any transformation.

###### Key words and phrases:

tropical geometry; algebraic differential equations; fundamental theorem;
Puiseux series

###### 2010 Mathematics Subject Classification:

Primary 14A20; Secondary 12H99, 13N99, 14T90, 14T99

##  1\. Introduction

Even though the first appearances of tropical methods in the area of
optimization can be traced back to the 1970’s, it is with the beginning of the
new millennium that these techniques have started to be applied more and more
often, and fruitfully, to geometry, eventually developing into an independent
and rich new branch of mathematics, with connections spreading as far as
computational biology and machine learning. The fundamental notion linking the
classical world and the tropical one is that of tropicalization: by means of a
valuation, this process moves problems of algebro-geometric nature to tropical
ones, the latter having an intrinsic combinatorial and polyhedral nature. A
comprehensive text to approach the subject is [MS21].

More recently, following the revival of interest around algebraic methods for
differential equations, and differential algebra in general, see e.g. [Rob14,
LH14, FMSW23], a tropical approach to the study of solutions to set of ODEs
have been introduced in [Gri17]. In this work the author gives a definition of
tropical differential equations and other objects needed to state a
fundamental theorem (similar to the fundamental theorem of tropical geometry,
see for example [MS21, Theorem 3.2.3]) in this context. This theorem was
successfully proven in [AGT16] and subsequently extended to the case of
partial differential equations in [FGLH+20], see [BFNS21] for a more
algorithmic description. Drawing inspiration from recent works where tropical
geometry is developed as geometry over the tropical semiring, such as [GG16,
MR18], a more general framework encompassing that of [Gri17] have been
proposed in [GM21]. This allows for a tropical theory of differential
equations to be developed also in the case of nontrivially valued fields of
coefficients and in this context a fundamental theorem for tropical
differential algebra, extending that of [AGT16], have been proven in [Mer23].
We will state our results in this language.

The fundamental theorem of tropical differential equations states equality
between the (weighted) support of formal power series solutions of a
differential ideal and the tropical solutions of the corresponding
tropicalized differential system over a valued uncountable algebraically
closed field of characteristic zero $K$, see Theorem 2.4. The direct extension
to more general formal series solutions such as formal Laurent series and
formal Puiseux series fails as it is presented in [FMSW23, Section 7]. In the
present work, we use transformations of the differential ideal and its
solution set to overcome this issue.

Transformations of differential equations and its solution sets are well
studied in the literature. Of particular interest have been diffeomorphisms
and homeomorphisms of dynamical systems and their solutions, the main subject
of differential topology. For non-autonomous and algebraic differential
equations, and their solutions, less is known. In these cases, transformations
are usually defined simultaneously on the coordinates and the functions; for
rational solutions see [FMSW23, Section 4] and references therein. In this
paper, we are loosely following the approach from [FMSW23] and use rational
transformations. More precisely, we will study rational maps
$\sigma(\mathbb{t},\mathbb{x})=(\mathbb{s}(\mathbb{t}),\mathbb{z}(\mathbb{t},\mathbb{x}))$
on the vector of independent variables $\mathbb{t}=(t_{1},\ldots,t_{m})$ and
differential indeterminates $\mathbb{x}=(x_{1},\ldots,x_{n})$. In order to
maintain the differential structure, we will have to perform the
transformations on the derivatives of the $\mathbb{x}$ accordingly. This will
give us a new differential system in $\mathbb{t}$ and $\mathbb{x}$. For
applying the transformation in the fundamental theorem, we additionally
require that

  1. (1)

$\mathbb{z}$ maps $(\mathbb{t},K[\\![\mathbb{t}]\\!]^{n})$ injectively to
$K[\\![\mathbb{t}]\\!]^{n}$;

  2. (2)

there has to be a tropical analogue of $\sigma$, i.e., there must be
$\Sigma(\mathbb{t},\mathbb{x})=(\mathbb{S}(\mathbb{t}),\mathbb{Z}(\mathbb{t},\mathbb{x}))$
acting on support sets $\mathbb{x}$ such that
$\mathbb{Z}(\mathbb{t},\textup{trop}(\varphi))=\textup{trop}(\mathbb{z}(\mathbb{t},\varphi))$
for $\varphi\in K[\\![\mathbb{t}]\\!]^{n}$;

  3. (3)

since we consider formal power series expanded around the origin, we require
that $\psi(0)=\mathbb{z}(\mathbb{t},\varphi(\mathbb{t}))|_{\mathbb{t}=0}$ is
defined and therefore assume that $0\in\mathrm{im}(\mathbb{s})$ such that
w.l.o.g. we set $\mathbb{s}(0)=0$; and

  4. (4)

since $\mathbb{z}(\mathbb{t},\mathbb{c})$ has to be defined for every constant
$\mathbb{c}\in K^{n}$, we assume that $\mathbb{z}(\mathbb{t},\mathbb{x})$ is
polynomial in $\mathbb{x}$.

For applying such transformations to univariate formal Laurent series and
formal Puiseux series, we use particular choices for $\sigma$, namely the
shift $\varphi\mapsto t^{n}\cdot\varphi$ and the power map $t\mapsto t^{n}$,
respectively. In the multivariate case, this transformation is a bit more
subtle. The actual computation for the value $n$ in the shift and the power
map can be found for concrete differential equations for instance by the
Newton polygon method for differential equations, see e.g. [ACJ03] and
references therein. Here we do not further study this finding and note that
there is no general algorithm known for finding minimal values $n$.

In Section 2, we first recall the classical formulations of the fundamental
theorem for tropical differential algebra. We then show the extension to
formal Laurent series solutions and formal Puiseux series solutions by
manipulating the given differential ideal in an appropriate way (see Theorem
2.8 for the ordinary case and Theorem 2.9 for the multivariate case). In
Section 3, we show that such transformations $\sigma$ cannot be used for
mapping transseries to formal power series. Thus, we can not generalize the
fundamental theorem to transseries solutions in this way. Moreover, we show
that one inclusion of the fundamental theorem still holds in the case of
transseries without any transformations. We conclude the paper by some
applications of the used transformations in the setting of approximation
theory.

##  2\. How to generalize the fundamental theorem to formal Puiseux series?

The fundamental theorem of tropical (partial) differential equations is
formulated and proven for (multi-variate) formal power series solutions of
systems of algebraic differential equations with formal power series
coefficients over an uncountable, algebraically closed fields of
characteristic zero $K$. In the ordinary case, where these series are
univariate, the coefficients of the given differential system may be
generalized to formal Laurent series. Essentially, this can be achieved by
multiplying with a monomial of lowest order occuring as a coefficient in the
system. Note that we may assume that the given system is a prime differential
ideal and finitely represented by a basis (see Ritt-Raudenbush basis theorem)
and thus, this order is indeed finite. Since the obtained system has again
formal power series coefficients, and their solution set is the same, the
fundamental theorem can be applied then. More generally, formal Puiseux series
coefficients of the given differential system can be transformed back to
formal power series with a power map in the independent variable.

In this document, we instead give a generalization of the solution set to
formal Puiseux series. The applied transformations, however, change the
differential system and also the solution set such that we can apply the
fundamental theorem for the case of formal power series solutions. Since this
transformation is one-to-one, we can recover the original system with its
formal Puiseux series solutions. We present a formulation of the fundamental
theorem by using inverse limits.

As we want to use the language of semirings to state our definitions and
results, here we make a quick recap about semirings and valuations. A
_semiring_ $(S,\oplus,\odot)$ is an algebraic structure satisfying all the
axioms to be a ring, but the existence of additive inverses. A semiring is
said to be _idempotent_ if for every $a\in S$ we have that $a\oplus a=a$.

######  Example 2.1.

  1. (1)

The set $\mathbb{B}=\\{0,\infty\\}$ equipped with operations $\oplus:=\min$
and $\odot:=+$ is an idempotent semiring, called the idempotent semiring of
Boolean numbers;

  2. (2)

The set $\mathbb{T}=\mathbb{R}\cup\\{\infty\\}$ equipped with operations
$\oplus:=\min$ and $\odot:=+$ is an idempotent semiring, called the idempotent
semiring of tropical numbers. It contains $\mathbb{B}$ as a subsemiring;

  3. (3)

For every $n\geq 1$ let $\mathbb{T}_{n}=\mathbb{R}^{n}\cup\\{\infty\\}$ with
$\oplus$ given by lexicographic minimum and $\odot:=+$. It is an idempotent
semiring, called the idempotent semiring of rank $n$ tropical numbers.

A sum of elements in an idempotent semiring is said to _tropically vanish_ if
by deleting any of the summands its result does not change. We adopt the
following generalised definition of valuation, as introduced and used in
[GG16, GM21, Mer23]: a _valuation_ on a ring $R$ is a map $v\colon\thinspace
R\to S$ to an idempotent semiring $S$ satisfying the following conditions for
all $a,b\in\mathbb{T}_{n}$:

  1. (1)

$v(0)=0_{S}$,

  2. (2)

$v(1)=v(-1)=1_{S}$,

  3. (3)

$v(ab)=v(a)\odot v(b)$,

  4. (4)

$v(a+b)\oplus v(a)\oplus v(b)$ tropically vanishes.

When $S=\mathbb{T}_{n}$ for some $n$, we will say that the valuation $v$ is of
_rank n_. Let us note that condition (4) is equivalent to saying that the
minimum of the terms $v(a+b),v(a),v(b)$ is achieved at least twice.

Let $K$ be an uncountable algebraically closed field of characteristic zero
equipped with a valuation $v_{K}\colon\thinspace K\to\mathbb{T}$. Let
$R_{m,n}=K[\\![t_{1},\ldots,t_{m}]\\!]\\{x_{1},\ldots,x_{n}\\}$ denote the
ring of differential polynomials in $x_{1},\ldots,x_{n}$ with formal power
series coefficients. Moreover, letting $\mathbb{t}:=(t_{1},\ldots,t_{m})$, we
write $\textup{Sol}_{A}$ for the set of solutions in a differential algebra
$A$ over $K[\\![\mathbb{t}]\\!]$. In the following, we will use multi-index
notations and operations such as multiplication, raising powers etc. are
performed component-wise.

For simplicity of treatment, here we will introduce the objects and
definitions appearing in the statement of the fundamental theorem of tropical
differential equations only in the case of ODEs (a treatment of the case of
PDEs can be found in [FGLH+20] when $v_{K}$ is the trivial valuation and in
[Mer22, Chapter 8] in full generality). In this case, the valuation $v_{K}$ on
$K$ induces a rank 2 valuation $v\colon\thinspace
K[\\![t]\\!]\to\mathbb{T}_{2}$ defined as

$A_{n_{0}}t^{n_{0}}+\dots\mapsto(n_{0},v_{K}(A_{n_{0}}))\in\mathbb{T}_{2}.$

When $v_{K}$ is the trivial valuation the image of this map is contained in a
subsemiring of $\mathbb{T}_{2}$ isomorphic to $\mathbb{T}$, thus we recover
the $t$-adic valuation used in [Gri17] and [AGT16].

Given a differential polynomial $f\in R_{1,n}$ of order less or equal
$r\in\mathbb{Z}_{\geq 0}$, let us write it as
$f(x)=\sum_{\lambda\in\Lambda}A_{\lambda}x^{\lambda}$ where $\Lambda$ is a
finite set of matrices in $\text{Mat}_{r\times n}(\mathbb{Z}_{\geq 0})$,
$A_{\lambda}$ is an element of $K[\\![t]\\!]$ for every $\lambda$ and
$x^{\lambda}$ is the differential monomial defined as
$\prod_{i,j}(x_{i}^{(j)})^{\lambda_{i,j}}$. The tropicalization
$\textup{trop}_{v}(f)$ of $f$ with respect to $v$ is defined as the polynomial
with coefficients in $\mathbb{T}_{2}$ obtained by applying $v$ coefficient-
wise to $f$:

$\textup{trop}_{v}(f):=\sum_{\lambda\in\Lambda}v(A_{\lambda})x^{\lambda}=\sum_{\lambda\in\Lambda}(\alpha_{\lambda}\text{,
}\beta_{\lambda})x^{\lambda}$

for the same $\Lambda$ and for $(\alpha_{\lambda}\text{,
}\beta_{\lambda})\in\mathbb{R}^{2}$ for every $\lambda\in\Lambda$. This
polynomial lives in an algebra of polynomials over $\mathbb{T}_{2}$ in
countably many variables $x_{i}^{(j)}$’s. Even though we could not talk
rigorously about elements of this algebra as tropical differential polynomials
over $\mathbb{T}_{2}$ (in fact this matter is quite subtle, see [GM21, Section
3]), we will nevertheless refer to them as tropical differential polynomials,
as this will not give rise to any ambiguity. Given a differential ideal
$G\subseteq R_{1,n}$ its tropicalization is the set
$\textup{trop}_{v}(G):=\\{\textup{trop}_{v}(f)\mid f\in G\\}$.

Given a power series $A\in K[\\![t]\\!]$, its tropicalization is performed by
applying $v_{K}$ coefficientwise, obtaining an element of
$\mathbb{T}[\\![t]\\!]$. We will denote this map as
$\widetilde{v}\colon\thinspace K[\\![t]\\!]\rightarrow\mathbb{T}[\\![t]\\!]$.
Lastly let
$\Phi\colon\thinspace\mathbb{T}[\\![t]\\!]\rightarrow\mathbb{T}_{2}$ be
defined as the homomorphism of semirings

$B_{n_{0}}t^{n_{0}}+\dots\mapsto(n_{0},B_{n_{0}})\in\mathbb{T}_{2}.$

######  Remark 2.2.

The valuation $v$, the homomorphism $\Phi$ and the map $\widetilde{v}$ fit
into a commutative triangle as follows:

(2.1)
${\mathbb{T}[\\![t]\\!]}$${K[\\![t]\\!]}$${\mathbb{T}_{2}.}$$\scriptstyle{\Phi}$$\scriptstyle{\widetilde{v}}$$\scriptstyle{v}$

This is an instance of _differential enhancement_ of the valuation $v$ (see
[GM21, Section 4]). Furthermore notice that the maps in the diagram above can
we defined in the same way considering Laurent or Puiseaux series instead of
power series, giving rise to an analogous commutative diagram as above. In
this case, though, the map $\Phi$ fails to be _reduced_ : this makes the
fundamental theorem of tropical differential equations fail in this case, for
similar reasons as explained in Example 3.1.

We finally introduce the notion of solution to a tropical differential
equation:

######  Definition 2.3.

Given a differential polynomial $f\in R_{1,n}$, a vector of tropical power
series $B=(B_{1},\dots,B_{n})\in\mathbb{T}[\\![t]\\!]^{n}$ is a _solution_ to
the tropicalization of $f$ if the expression

$\textup{trop}_{v}(f)(B):=\bigoplus_{\lambda\in\Lambda}v(A_{\lambda})\bigodot_{i,j}(\Phi(d^{j}(B_{i})))^{\odot{\lambda_{i,j}}}\in\mathbb{T}_{2}$

tropically vanishes. I.e. if when plugging in $\Phi(d^{j}(B_{i}))$ for
$x_{i}^{(j)}$, the minimum is achieved at least twice in $\mathbb{T}_{2}$.

As here we will not deal with the fully general case as introduced in [GM21,
Section 4], and deal only with the case of the homomorphism $\Phi$, we will
use the notation $\textup{Sol}_{\mathbb{T}[\\![t]\\!]}(\textup{trop}_{v}(f))$
for the set of solutions to the tropicalization of the differential equation
$f$ in $\mathbb{T}[\\![t]\\!]^{n}$, as introduced in Definition 2.3 above. We
have the following results:

######  Theorem 2.4 ([AGT16, FGLH+20]).

Let $K$ be an uncountable, algebraically closed field of characteristic zero,
and let $v_{K}$ be the trivial valuation. Let $G$ be a differential ideal in
the ring $R_{m,n}$. Then

$\textup{trop}_{\widetilde{v}}(\textup{Sol}_{K[\\![\mathbb{t}]\\!]}(G))=\textup{Sol}_{\mathbb{T}[\\![\mathbb{t}]\\!]}(\textup{trop}_{v}(G)).$

In the case of ordinary differential equations the following generalisation
holds:

######  Theorem 2.5 ([Mer23]).

Let $K$ be an uncountable, algebraically closed field of characteristic zero,
equipped with a valuation $v_{K}\colon\thinspace K\rightarrow\mathbb{T}$. Let
$G$ be a differential ideal in the ring $R_{1,n}$. Then

$\textup{trop}_{\widetilde{v}}(\textup{Sol}_{K[\\![\mathbb{t}]\\!]}(G))=\textup{Sol}_{\mathbb{T}[\\![\mathbb{t}]\\!]}(\textup{trop}_{v}(G)).$

In the second case, we suspect that an analogous theorem for PDEs should hold.
To simplify the treatment in the following, we will focus on the trivially
valued case from here on, this way we will be able to treat both the case of
ODEs and PDEs simultaneously.

###  2.1. Ordinary case

Throughout the section, let $G$ be a differential ideal in the ring $R_{1,n}$.

#### Laurent series

Let us look for formal Laurent series solutions of $G$ instead of formal power
series solutions. A univariate formal Laurent series $\varphi=\sum_{i\geq
0}c_{i}t^{i-l}$ with $c_{i}\in K$, $l\in\mathbb{Z}$, can be transformed to the
formal power series $t^{l}\cdot\varphi(t)=\sum_{i\geq 0}c_{i}t^{i}$ by the
transformation $\sigma_{l}(t,x)=(t,t^{l}\,x)$. Let us note that $l$ could be
chosen as positive number, because otherwise the given Laurent series
$\varphi$ is already a formal power series. In order to keep the original and
transformed system distinct, we will write $z=t^{l}\,x$. For the derivatives
we obtain that

$z^{\prime}=lt^{l-1}\cdot x+t^{l}\cdot x^{\prime},\,\text{ etc. }$

Let us define
$\sigma_{l}(t,x_{1},\ldots,x_{n})=(t,t^{l}\,x_{1},\ldots,t^{l}\,x_{n})$. In
$G$, we thus replace $x_{j}$ by $z_{j}:=t^{l}\,x_{j}$ and the derivatives by
$x_{j}^{\prime}=\frac{z_{j}^{\prime}-lt^{l-1}z_{j}}{t^{l}}$, and so on. By
taking the numerators, we obtain a system in
$K[\\![t]\\!]\\{z_{1},\ldots,z_{n}\\}$, denoted by $G^{(l)}$. Moreover, a
formal Laurent series $\varphi=(\varphi_{1},\ldots,\varphi_{n})$, where all
$\varphi_{j}$ have order greater or equal to $l$, is a solution of $G$ if and
only if
$(\psi_{1},\ldots,\psi_{n}):=(t^{l}\,\varphi_{1},\ldots,t^{l}\,\varphi_{n})$
is a formal power series solution of $G^{(l)}$.

#### Puiseux series

Let $\varphi=\sum_{i\geq 0}c_{i}t^{(i-l)/k}$ be a Puiseux series where
$c_{i}\in K$, $l\in\mathbb{Z}$, and $k>0$ is the ramification index, i.e., the
minimal number such that $\varphi(t^{k})$ is a formal Laurent series. Set
$\sigma_{k,l}(t,x)=(t^{k},t^{l}\,x)$. Then $\psi:=t^{l}\,\varphi(t^{k})$ is a
formal power series in $t$ and we transform the derivatives as
$x^{\prime}=kt^{k-1+l}\cdot z^{\prime}+lt^{l-1}\cdot z$, etc. In $G$, we thus
replace $t$ by $t^{k}$, $x_{j}$ by $t^{l}\cdot z_{j}$ and the derivatives
$x_{j}^{\prime}=\frac{z_{j}^{\prime}-lt^{l-1}z_{j}}{kt^{k-1+l}}$, etc. After
considering just the numerators, we obtain a system $G^{(k,l)}\in R_{1,n}$
such that $(\varphi_{1},\ldots,\varphi_{n})$ is a formal Puiseux series
solution for $G$ if and only if
$(\psi_{1},\ldots,\psi_{n}):=(t^{l}\,\varphi_{1}(t^{k}),\ldots,t^{l}\,\varphi_{n}(t^{k}))$
is a formal power series solution for $G^{(k,l)}$.

#### Fundamental theorem

Let us now connect the above observations to the fundamental theorem. We will
work with radical differential ideals $G$ because the solution set of any
differential ideal and its radical coincide. For a given set of differential
polynomials $\mathcal{F}$, we denote by $[\mathcal{F}]$ the smallest radical
differential ideal containing $\mathcal{F}$. According to Ritt-Raudenbush
theorem [BH19], every radical differential ideal is finitely generated. Let us
now show that the generators of a radical differential ideal transform
properly under the above transformation.

######  Lemma 2.6.

Let $G$ be the radical differential ideal generated by $F_{1},\ldots,F_{N}\in
R_{1,n}$, i.e. $G=[F_{1},\ldots,F_{N}]$. Then
$G^{(k,l)}=[F_{1}^{(k,l)},\ldots,F_{N}^{(k,l)}]$.

###### Proof.

For the proof, we use characteristic sets representing the prime components of
a given radical differential ideal [Kol73]. Since the order, leaders, initials
and separants of $G$ and $G^{(k,l)}$ coincide (up to multiplication with
monomials in $K[t]$ for the latter two), the characteristic sets of the prime
components of $G$ and $G^{(k,l)}$, denoted by $\Sigma$ and $\Sigma^{(k,l)}$
respectively, are transformed into each other. ∎

In the following, let us denote by $K(\\!(t)\\!)=K[t^{-1}][\\![t]\\!]$ and
$K\\{\\!\\{t\\}\\!\\}=\bigcup_{k\in\mathbb{Z}_{>0}}K(\\!(t^{1/k})\\!)$ the
field of formal Laurent series and the field of formal Puiseux series,
respectively. Since the transformations above change the support of the
solutions, namely by a shift $l$ or a multiplication with $\tfrac{1}{k}$, we
also have to include this in the solutions of the tropicalized differential
systems. Then the transformations above can be written as follows.

######  Proposition 2.7.

Let $K$ be a field of characteristic zero, equipped with a valuation
$v_{K}\colon\thinspace K\rightarrow\mathbb{T}$ and $G\subset R_{1,n}$ be a
differential ideal. With notations as in diagram 2.1 for Puiseaux series over
$K$, the following holds:

$\textup{trop}_{\widetilde{v}}(\textup{Sol}_{K\\{\\!\\{t\\}\\!\\}}(G))=\bigcup_{l,k\in\mathbb{N}}\\{t^{-l}A(t^{k})\mid
A(t)\in\textup{trop}_{\widetilde{v}}(\textup{Sol}_{K[\\![t]\\!]}(G^{(k,l)}))\\}.$

###### Proof.

From the definition of $G^{(k,l)}$, as we have seen above, it follows that
every formal Puiseux series solution of $G$ transforms into a formal power
series solution of $G^{(k,l)}$ for some $k,l\in\mathbb{Z}_{>0}$.

For the converse direction, let $\psi=(\psi_{1},\ldots,\psi_{n})$ be a formal
power series solution for some $G^{(k,l)}$ with $k,l\in\mathbb{Z}_{>0}$. Then
the formal inverse $\sigma_{k,l}^{-1}(t,z):=(t^{1/k},z/t^{l})$ transforms
every equation of $G^{(k,l)}$ back to an equation in $G$, where we might have
to multiply with elements in $K[\\![t]\\!]$. Since $\sigma_{k,l}^{-1}(t,\psi)$
defines a formal Puiseux series solution $\varphi$ of the transformed
equations and, by Lemma 2.6, they generate $G$, $\varphi$ is a formal Puiseux
series solution of $G$. ∎

In Proposition 2.7, for the right hand side, the fundamental theorem for
formal power series, Theorem 2.4 and 2.5 can be applied so that we obtain the
following generalization:

######  Theorem 2.8.

Let $K$ be an uncountable, algebraically closed field of characteristic zero,
equipped with a valuation $v_{K}\colon\thinspace K\rightarrow\mathbb{T}$. Let
$G\subset R_{1,n}$ be a differential ideal. Then

$\textup{trop}_{\widetilde{v}}(\textup{Sol}_{K\\{\\!\\{t\\}\\!\\}}(G))=\bigcup_{l,k\in\mathbb{N}}\\{t^{-l}A(t^{k})\mid
A(t)\in\textup{Sol}_{\mathbb{T}[\\![t]\\!]}(\textup{trop}(G^{(k,l)}))\\}.$

###  2.2. Partial case

Let us now generalize the previous results to the multivariate case of
$R_{m,n}$ with $m>1$, when the field of coefficients $K$ is equipped with the
trivial valuation. For this purpose, let us introduce the field of formal
Puiseux series in several variables as follows.

A convex rational polyhedral cone is a subset of $\mathbb{R}^{m}$ of the form

$C=\\{\lambda_{1}v_{1}+\cdots+\lambda_{r}v_{r}\mid\lambda_{i}\in\mathbb{R},\lambda_{i}\geq
0\\},$

where $v_{1},\ldots,v_{r}\in\mathbb{Q}^{m}$ are vectors. A cone is said to be
strongly convex if it contains no nontrivial linear subspaces. Let $C$ be a
strongly convex rational polyhedral cone and let
$\mathbb{d}=(d_{1},\ldots,d_{m})\in\mathbb{Z}_{>0}^{m}$. Set
$C_{\mathbb{d}}=C\cap(\frac{1}{d_{1}}\cdot\mathbb{Z},\ldots,\frac{1}{d_{m}}\cdot\mathbb{Z})$.
Using multi-index notation, the set of formal sums $\sum_{\mu\in
C_{\mathbb{d}}}a_{\mu}t^{\mu}$, where $a_{\mu}\in K$, forms a ring and is
denoted by $K[\\![C_{\mathbb{k}}]\\!]$. Let $\mathbb{w}\in\mathbb{R}_{>0}^{m}$
have rationally independent entries over $\mathbb{Q}$. Let $H_{\mathbb{w}}$ be
the half-space
$\\{\mathbb{t}\in\mathbb{R}^{m}\mid\mathbb{w}\cdot\mathbb{t}\geq 0\\}$. The
ring of power series with support in a convex rational polyhedral cone in
$H_{\mathbb{w}}$ and fractional exponents with denominators
$\mathbb{d}\in\mathbb{Z}_{>0}^{m}$ is defined by

$K[\\![\mathbb{t}]\\!]_{\mathbb{d}}^{\mathbb{w}}=\bigcup_{C\subseteq
H_{\mathbb{w}}}K[\\![C_{\mathbb{d}}]\\!].$

By allowing the support in some translate of $C$ we obtain the field of
(multivariate-) formal Laurent series

$K(\\!(\mathbb{t})\\!)_{\mathbb{d}}^{\mathbb{w}}=\bigcup_{\mathbb{l}\in(\frac{1}{d_{1}}\cdot\mathbb{Z},\ldots,\frac{1}{d_{m}}\cdot\mathbb{Z})}\mathbb{t}^{\mathbb{l}}\cdot
K[\\![\mathbb{t}]\\!]^{\mathbb{w}}$

and the field of (multivariate) formal Puiseux series

$K(\\!(\mathbb{t})\\!)_{*}^{\mathbb{w}}=\bigcup_{\mathbb{d}\in\mathbb{Z}_{>0}^{m}}K(\\!(\mathbb{t})\\!)_{\mathbb{d}}^{\mathbb{w}}.$

Let us note that a different choice of $\mathbb{w}$ leads to a different
representation of the Puiseux series. For a given system of differential
equations, their Puiseux series solutions can thus be translated one-to-one
for different $\mathbb{w}$’s.

Following [ACJ03, Section 8], a formal Puiseux series $\varphi$ with support
in a cone $\mathbb{l}+C_{\mathbb{d}}$ can be transformed into a formal power
series with exponents in the first quadrant. More precisely, there exists a
rational map $\kappa(\mathbb{t})$ with $\kappa(0)=0$,
$\varphi(\kappa(\mathbb{t}))\in K[\\![\mathbb{t}]\\!]$ and formally invertible
as a vector of Laurent monomials with differentiable inverse, found as a
finite composition of “combinatorial blow-ups”, i.e. transformations

$\sigma_{I,j}(t_{1},\ldots,t_{m})=(s_{1},\ldots,s_{m})~{}\text{ with
}~{}s_{i}=\begin{cases}t_{i}\,t_{j}&i\in I,i\neq j\\\
t_{i}&\textit{otherwise}\end{cases}$

where $I\subset\\{1,\ldots,m\\}$ contains at least two elements and $j\in I$.
Thus, eventually we have that

$\kappa(\mathbb{t})=(t_{1}\cdot\mathbb{t}^{\alpha_{1}},\ldots,t_{m}\cdot\mathbb{t}^{\alpha_{m}})$

for some $\alpha_{i}\in\mathbb{Z}_{\geq 0}^{m}$ where we use the notation
$\mathbb{t}^{\beta}=t_{1}^{\beta_{1}}\cdots t_{m}^{\beta_{m}}$. Let us denote
by $\mathcal{B}$ the set of all such maps $\kappa$ that are a finite
composition of combinatorial blow-ups. Additionally, we perform the shift by
the translation vector $\mathbb{l}$ such that we obtain the transformation
$\sigma_{\mathbb{d},\mathbb{l}}(\mathbb{t},\mathbb{x})=(\kappa(\mathbb{t}),\mathbb{t}^{\mathbb{l}}\,\mathbb{x})=:(\mathbb{s},\mathbb{z})$
for
$\kappa\in\mathcal{B},\mathbb{l}\in(\frac{1}{d_{1}}\cdot\mathbb{Z},\ldots,\frac{1}{d_{m}}\cdot\mathbb{Z})$
where $\kappa$ is depending on $\mathbb{w}$ and $\mathbb{d}$.

Notice that the maps $\sigma_{\mathbb{d},\mathbb{l}}(\mathbb{t},\mathbb{x})$
and therefore $\kappa$ can be defined analogously on $\mathbb{B}[\\![t]\\!]$
as all the operations in their definition involve transformations of the
monomials of the power series. Also in this case we will denote them with the
same symbols. Thus, we obtain

$\mathrm{supp}(\varphi_{i}(\kappa(\mathbb{t})))=\kappa(\mathrm{supp}(\varphi_{i})).$

Let $G$ be a differential ideal. Similarly to the ordinary case, we apply the
transformation $\sigma_{\mathbb{k},\mathbb{l}}(\mathbb{t},\mathbb{x})$ to
$\mathbb{t}$, $\mathbb{x}$ and the partial derivatives of the components of
$\mathbb{x}$ accordingly, namely by
$\mathbb{x}^{\prime}=\mathbb{t}^{-l-1}\,\kappa^{\prime-1}\cdot(\mathbb{t}\,\mathbb{z}^{\prime}+l\,\mathbb{z})$
where the derivatives denote the Jacobi matrices w.r.t. $\mathbb{t}$. Let us
note that $\kappa^{\prime}$ is invertible and $\kappa^{\prime}$ and
$\kappa^{\prime-1}$ contain only monomials in $\mathbb{Z}[\mathbb{t}]$. Then a
new system in $\mathbb{t}$ and $\mathbb{z}$ is obtained and we denote it,
after cancelling $\mathbb{t}^{l}$ and multiplying with the common denominator
in $K[\\![\mathbb{t}]\\!]$, by $G^{(\mathbb{k},\mathbb{l})}$.

For applying the fundamental theorem, we have to actually compute the formal
inverse of the transformation $\sigma_{\mathbb{k},\mathbb{l}}$. The
transformations $\sigma_{I,j}$ have the formal inverse

$\sigma_{I,j}^{-1}(\mathbb{s})=\mathbb{t}~{}\text{ with
}~{}t_{i}=\begin{cases}s_{i}/s_{j},&i\in I,i\neq j\\\ s_{i}\end{cases}.$

Thus, we obtain that $\kappa^{-1}$ is the finite decomposition of the
$\sigma_{I,j}^{-1}$ and of the form

$\kappa^{-1}(\mathbb{s})=(s_{1}\cdot\mathbb{s}^{\beta_{1}},\ldots,s_{m}\cdot\mathbb{s}^{\beta_{m}})$

for some $\beta_{1},\ldots,\beta_{m}\in\mathbb{Z}^{m}$ and
$\sigma_{\mathbb{k},\mathbb{l}}^{-1}(\mathbb{s},\mathbb{z})=(\kappa^{-1}(\mathbb{s}),\mathbb{s}^{-l}\cdot\mathbb{z})$.
Since $\kappa$ is invertible such that the inverse again commutes with taking
supports, we obtain

(2.2)
$\textup{trop}_{\widetilde{v}}(\varphi_{i})=\kappa^{-1}(\textup{trop}_{\widetilde{v}}(\varphi_{i}(\kappa(\mathbb{t})))).$

Lemma 2.6 and Proposition 2.7 can be generalized to the multivariate case by
simply replacing $R_{1,n}$ with $R_{m,n}$ and every $k,l$ with
$\mathbb{d},\mathbb{l}$ in the statements and its proofs. Then, using the
above transformation, in the case of trivial valuation, the fundamental
theorem 2.8 generalizes to the multi-variate case in the following sense.

######  Theorem 2.9.

Let $K$ be an uncountable, algebraically closed field of characteristic zero.
Let $G\subset R_{m,n}$ be a differential ideal and let
$\mathbb{w}\in\mathbb{R}_{>0}^{m}$ have rationally independent entries over
$\mathbb{Q}$. Then

$\textup{trop}_{\widetilde{v}}(\textup{Sol}_{K(\\!(\mathbb{t})\\!)_{*}^{\mathbb{w}}}(G))=\bigcup_{\mathbb{d},\mathbb{l}}\\{\sigma_{\mathbb{d},\mathbb{l}}^{-1}(A)\mid
A(\mathbb{t})\in\textup{Sol}_{\mathbb{T}[\\![\mathbb{t}]\\!]}(\textup{trop}(G^{(\mathbb{d},l)}))\\}.$

where the union is taken over all $\mathbb{d}\in\mathbb{Z}_{>0}^{m}$ and
$\mathbb{l}\in(\frac{1}{d_{1}}\cdot\mathbb{Z},\ldots,\frac{1}{d_{m}}\cdot\mathbb{Z})$.

######  Example 2.10.

Let us consider the differential ideal generated by the differential
polynomials

$F_{1}=t_{1}\,\frac{\partial\,x}{\partial
t_{1}}+t_{2}\,\frac{\partial\,x}{\partial
t_{2}}-x/2,\,\,\,F_{2}=\frac{\partial\,x}{\partial
t_{1}}-\frac{\partial\,x}{\partial t_{2}}$

inside $\mathbb{C}[\\![t_{1},t_{2}]\\!]\\{x\\}$. By tropicalizing

$\frac{\partial^{i+j}}{\partial t_{1}^{i}\partial
t_{2}^{j}}F_{1}=t_{1}\,\frac{\partial^{i+j+1}\,x}{\partial t_{1}^{i+1}\partial
t_{2}^{j}}+t_{2}\,\frac{\partial^{i+j+1}\,x}{\partial t_{1}^{i}\partial
t_{2}^{j+1}}+(i+j-1/2)\frac{\partial^{i+j}\,x}{\partial t_{1}^{i}\partial
t_{2}^{j}},$

we directly see that $\varphi=0$ is the only possible formal power series
solution. Let us consider for $\mathbb{d}=(1,1/2),\mathbb{l}=(0,-1/2)$ (and
$\kappa(t_{1},t_{2})=(t_{1}t_{2},t_{2})$) the transformation
$\sigma_{\mathbb{d},\mathbb{l}}(\mathbb{t},x)=((t_{1}t_{2},t_{2}),t_{2}^{-1/2}\cdot
x)$ such that

$\frac{\partial\,x}{\partial
t_{1}}=\frac{1}{\sqrt{t_{2}}}\,\frac{\partial\,z}{\partial
t_{1}},\,\frac{\partial\,x}{\partial
t_{2}}=\sqrt{t_{2}}\,\frac{\partial\,z}{\partial
t_{2}}-\frac{t_{1}}{\sqrt{t_{2}}}\,\frac{\partial\,z}{\partial
t_{2}}+\frac{1}{2\sqrt{t_{2}}}\,z$

leads to the differential system generated by

$F_{1}^{(\mathbb{k},\mathbb{l})}=\frac{\partial\,z}{\partial
t_{2}},\,\,\,F_{2}^{(\mathbb{k},\mathbb{l})}=(1+t_{1})\,\frac{\partial\,z}{\partial
t_{1}}-t_{2}\,\frac{\partial\,z}{\partial t_{2}}-z/2.$

For tropical solutions $A\in\mathbb{B}[\\![t_{1},t_{2}]\\!]$ of
$\textup{trop}_{v}(F_{1}^{(\mathbb{k},\mathbb{l})})$ we see that $A$ has to
belong to $\mathbb{B}[\\![t_{1}]\\!]$. Similarly, by considering
$\textup{trop}_{v}(F_{2}^{(\mathbb{k},\mathbb{l})})$ and all its derivatives
with respect to $t_{1}$, a tropical solution for this system has to belong to
$\mathbb{B}[\\![t_{1}]\\!]$. Indeed, the formal Puiseux series solution
$\varphi=c\cdot\sqrt{t_{1}+t_{2}}=\sum_{i\geq 0}\binom{1/2}{i}\cdot
t_{1}^{i}t_{2}^{1/2-i}$ of $[F_{1},F_{2}]$ transforms to the formal power
series solution $\psi=\sum_{i\geq 0}\binom{1/2}{i}\cdot t_{1}^{i}$ of
$[F_{1}^{(\mathbb{k},\mathbb{l})},F_{2}^{(\mathbb{k},\mathbb{l})}]$ which has
exactly the prescribed support set, and

$\textup{trop}_{\widetilde{v}}(\varphi)=\sigma_{\mathbb{d},\mathbb{l}}^{-1}(\textup{trop}_{\widetilde{v}}(\psi))=\left\\{\begin{pmatrix}-1/2\\\
1/2\end{pmatrix},\begin{pmatrix}1/2\\\ -1/2\end{pmatrix},\ldots\right\\}$

is a solution of $\textup{trop}([F_{1},F_{2}])$.

##  3\. Can we generalize the fundamental theorem to series involving
logarithms?

Let us now consider series which might involve logarithms. For this purpose we
will use (grid-based) transseries, which generalize formal Puiseux series
(over $\mathbb{R}$). Informally speaking, transseries are formal Hahn series
of real powers of the indeterminate $t$, exponentials, logarithms and their
compositions, with real coefficients. It is required that the numbers of
iterations of occurring exponentials and logarithms is finite and that the
series are well-based. For a rigorous definition and details on transseries
see [Edg09] or [VdH06]. A generalization to series with complex coefficients
is presented in [vdH01]. These series generalize formal Puiseux series over
$\mathbb{C}$.

Let $(\mathcal{M},\cdot)$ denote a totally ordered monomial group with powers
in a real trigonometric field $K$ such as $\mathbb{R}$ (or $K$ to be the
complexification of a real trigonometric function field such as $\mathbb{C}$).
We will use $\mathcal{M}$ as the product and composition of the variables
$\mathbb{t}$, logarithms and exponentials in $\mathbb{t}$. A grid-based
transseries is then a series $\varphi=\sum_{m\in\mathcal{M}}\varphi_{m}\cdot
m$ such that the support
$\mathrm{supp}(\varphi)=\\{m\in\mathcal{M}\mid\varphi_{m}\neq 0\\}$ is
finitely generated in $\mathcal{M}$. We will denote as
$K[\\![\mathcal{M}]\\!]$ the field of grid-based transseries over
$\mathcal{M}$ with coefficients in $K$.

Let $d\colon\thinspace\mathcal{M}\rightarrow K[\\![\mathcal{M}]\\!]$ be a map
satisfying the Leibniz rule and such that $d(\log(m))=d(m)/m$ for all
$m\in\mathcal{M}$. Then, by [VdH06, Theorem 5.1], the map $d$ extends to an
exp-log derivation on $K[\\![\mathcal{M}]\\!]$, thus making
$(K[\\![\mathcal{M}]\\!],d)$ into a differential field.

Denoting with $\leq$ the total order on $\mathcal{M}$, we can regard
$\mathcal{M}\cup\\{\infty\\}$ as a semifield: we endow $\mathcal{M}$ with
addition given as $\oplus:=\min_{\leq}$ and with multiplication given by the
group operation of $\mathcal{M}$. These two operations are extended to
$\mathcal{M}\cup\\{\infty\\}$ by $m\oplus\infty=m$ and $m\cdot\infty=\infty$
for all $m\in\mathcal{M}$. We equip $K[\\![\mathcal{M}]\\!]$ with a valuation
$v\colon\thinspace K[\\![\mathcal{M}]\\!]\rightarrow\mathcal{M}$ in a
tautological way by sending a power series $\varphi$ as above to the minimum
of its support:

$v(\varphi):=\min_{\leq}\mathrm{supp}(\varphi).$

It is straightforward to check that the map $v$ is indeed a valuation.

Considering the idempotent semiring $\mathbb{B}[\\![\mathcal{M}]\\!]$ of
Boolean transseries, we can make it into a differential semiring with a
differential $d$ defined analogously as above. The map
$\Phi\colon\thinspace(\mathbb{B}[\\![\mathcal{M}]\\!],d)\rightarrow\mathcal{M}$
sending a boolean transseries to the minimum of its support in $\mathcal{M}$
is a tropical pair $\mathbf{S}$, in the language of [GM21]. It is non-reduced
(indeed its restriction to $\mathbb{B}(\\!(t)\\!)$ inside
$\mathbb{B}[\\![\mathcal{M}]\\!]$ is the non-reduced tropical pair
$\mathbb{B}(\\!(t)\\!)\rightarrow\mathbb{Z}\cup\\{\infty\\}$).

Finally, we can define a map $\widetilde{v}\colon\thinspace
K[\\![\mathcal{M}]\\!]\rightarrow\mathbb{B}[\\![\mathcal{M}]\\!]$ by
coefficientwise application of the trivial valuation on the elements of
$K[\\![\mathcal{M}]\\!]$, i.e. sending a transseries with coefficients in $K$
to its support. By definition of this map and of the differentials of its
domain and codomain, it is clear that it commutes with the differentials. In
conclusion, we built the following commutative diagram:

${\mathbb{B}[\\![\mathcal{M}]\\!]}$${K[\\![\mathcal{M}]\\!]}$${\mathcal{M}}$$\scriptstyle{\Phi}$$\scriptstyle{\widetilde{v}}$$\scriptstyle{v}$

which satisfies all the hypothesis to be a differential enhancement of the
valuation $v$, but the hypothesis of $\Phi$ being reduced. Let us denote as
$\mathbf{v}=(v,\widetilde{v})\colon\thinspace(K[\\![\mathcal{M}]\\!],d)\to\mathbf{S}$
the differential enhancement above. Given a differential ideal $I\subset
K[\\![\mathcal{M}]\\!]\\{x_{1},\ldots,x_{n}\\}$, with the same proof as in
[GM21, Proposition 5.2.2], even if the pair $\mathbf{S}$ is non-reduced, we
have that the following inclusion holds:

$\textup{trop}_{\widetilde{v}}(\textup{Sol}_{K[\\![\mathcal{M}]\\!]}(I))\subseteq\textup{Sol}_{\mathbf{S}}(\textup{trop}_{v}(I)).$

Now, let us give an example in which the equality of the two sets above does
not hold. Equivalently, the fundamental theorem does not hold for the
differential enhancement $\mathbf{v}$ introduced above.

######  Example 3.1.

Consider the differential polynomial
$F=tx^{\prime}-x-t\in\mathbb{C}[\\![t]\\!]\\{x\\}$. The differential ideal
generated by $F$ is prime. The solutions to the algebraic differential
equation $F=0$, and thus to $[F]$, are of the form $\varphi=ct+t\log
t\in\mathbb{C}[\\![t,\log(t)]\\!]$, where $c\in\mathbb{C}$. Thus,

$\textup{trop}_{\widetilde{v}}(\textup{Sol}(F))=\\{t+t\log t,t\log t\\}.$

We can see the differential ring $\mathbb{C}[\\![t,\log(t)]\\!]$ as a
differential ring $\mathbb{C}[\\![\mathcal{M}]\\!]$ for the totally ordered
group generated by the symbols $t$ and $\log(t)$, with order given by
$t\geq\log(t)$. The derivation sends $\log(t)$ to $t^{-1}$. As a semiring,
$\mathcal{M}\cup\\{\infty\\}$ is isomorphic to
$\mathbb{Z}^{2}\cup\\{\infty\\}\subset\mathbb{T}_{2}$, thus we can write an
element $t^{n}\log(t)^{m}\in\mathcal{M}$ as $(n,m)$ and we consider
$\mathbb{T}_{2}$ to be the target of the valuation $v$.

Let us now compute the tropical differential polynomials
$\textup{trop}(\\{F\\})$ and their solutions. From
$\textup{trop}_{v}(dF)=(1,0)x^{\prime\prime}+(0,0)$ we know that there is no
solution of $\textup{trop}_{v}(\\{F\\})$ in $\mathbb{B}[\\![t]\\!]$. Since the
fundamental theorem applies to solutions in $\mathbb{C}[\\![t]\\!]$, this is
also telling us that there are no formal power series solutions of $[F]$.

Looking for solutions $A\in\mathbb{B}[\\![\mathcal{M}]\\!]$ to
$\textup{trop}_{v}(F)=(1,0)x^{\prime}+(0,0)x+(1,0)$ we get that a solution is
either of the form $A=t\log(t)+\dotsb$ or of the form $A=t+t\log(t)+\dotsb$.
Taking further derivatives of $F$, we obtain

$\textup{trop}_{v}(d^{k}F)=(1,0)x^{(k+1)}+(0,0)x^{(k)}$

and evaluating $\textup{trop}_{v}(d^{k}F)$ in $A$, we obtain no further
condition on the tropical solutions. We can now prove that the fundamental
theorem does not hold in this context. More precisely, that given a classical
solution $\varphi$, any boolean transseries of the form
$A=\textup{trop}_{\widetilde{v}}(\varphi)+B$ with
$B\in\mathbb{B}[\\![\mathcal{M}]\\!]$, $\Phi(B)>t\log t$ is in
$\textup{Sol}(\textup{trop}_{v}([F]))$.

Let us prove that $A:=\textup{trop}_{\widetilde{v}}(\varphi)+B$ is a solution
for $\textup{trop}_{v}([F])$. Indeed, for every $k\in\mathbb{N}$, the
following equality holds:

$\Phi(d^{k}A)=\Phi(d^{k}\textup{trop}_{\widetilde{v}}(\varphi))$

and, as $\textup{trop}_{\widetilde{v}}(\varphi)$ is in
$\textup{Sol}(\textup{trop}_{v}([F]))$, by definition of a tropical solution,
every boolean transseries of the form
$A=\textup{trop}_{\widetilde{v}}(\varphi)+B$ is in
$\textup{Sol}(\textup{trop}_{v}([F]))$ as well.

######  Remark 3.2.

Notice that, ultimately, the fundamental theorem does not hold in this context
(and analogously it does not hold when looking for solutions in
$\mathbb{B}(\\!(t)\\!)$) as an essential feature of the derivation over power
series is that 0 is a sink. Conversely, when looking at elements like $t^{-1}$
there is no $k\in\mathbb{N}$ such that $d^{k}t^{-1}=0$. Even the non-
reducedeness of the pair $\Phi$ does not play a role, as the same problem
would arise by considering its reduction.

#### Transformations

Let us explain why no general transformation from transseries to formal power
series, mapping the given system of algebraic differential equations to
another system of algebraic differential equations fulfilling (almost all of)
our assumptions in the introduction, exists. For this purpose, let us consider
the simple case where transseries solutions $\varphi=\sum_{i\geq
0}c_{i}(\log(t))t^{i}\in K[\log(t)][\\![t]\\!]$ of $G\subset R_{1,n}$ are
sought. Let us notice that we could also study $K[t][\\![\exp(t)]\\!]$ instead
because every transseries involving only nested exponentials and logarithms
can be written as a logarithm-free transseries by simply using the
transformation $s(t)=\exp^{h}(t)$ where $\log^{h}(t)$ is the logarithmic
depth.

First, $\sigma$ cannot be chosen to be rational because otherwise expressions
such as $\varphi=\log(t)$ cannot be transformed into a formal power series. We
might allow more general type of transformations as rational functions, but
want to keep the other hypothesis (1), (2), (3) from the introduction. Then
the natural choice for $\varphi=\log(t)$, namely $s(t)=\exp(t)$, is excluded
by the condition $s(0)=0$. Also any other formal power series does not work,
because for $s(t)=t\cdot s_{0}(t)$ with $s_{0}(t)\in K[\\![t]\\!]$, we obtain
$\psi=\log(s(t))=\log(t)+\log(s_{0}(t))$ which is not a formal power series
since $\psi(0)$ is undefined. Dropping the assumption that $s(0)=0$ is
problematic, because then $\varphi(s(t))$ might not be evaluated at $t=0$. We
leave it as an open question to find $z(t,x)$ such that this issue is solved.

Second, when we use $s(t)=t$ and consider just transformations in $x$, for
every constant $\varphi=c$ the evaluation $z(t,\varphi)$ has to be defined. If
we allow more general transformations than rational functions, the natural
choice would be $z(t,x)=\exp(x/t^{k})$ for $k\in\mathbb{Z}$. In this way,
$\varphi=t^{k}\log(t)$ would be mapped to a formal power series. Then,
however, for $\varphi=\log(t)(t+1)$ we obtain that $z(t,\varphi)=t^{t+1}$ (for
$k=0$) or $z(t,\varphi)=t^{t-1}$ (for $k=1$) which are not formal power
series. We believe that there will not be any transformation $z$ mapping all
transseries $K[\log(t)][\\![t]\\!]$ to formal power series.

##  4\. Other applications

The transformations used for formal Laurent series and formal Puiseux series,
respectively, can also be applied in other settings than that of tropical
differential algebra. One other application, which relates to the fundamental
theorem of tropical differential algebra, is given in approximation theory
leading to the following result (cf. [DL84, Theorem 3.1]).

######  Theorem 4.1.

Let $K\in\\{\mathbb{C},\mathbb{R},\mathbb{Q}_{p}\\}$. Let $G$ be a finite
differential system in $\mathbb{Q}[t]\\{x_{1},\ldots,x_{n}\\}$ and let
$l\in\mathbb{Z}$ and $k\in\mathbb{Z}_{>0}$. Then there is an algorithm for
deciding whether there is a solution $(\varphi_{1},\ldots,\varphi_{n})\in
K(\\!(t^{1/k})\\!)^{n}$ of $G$ with $\mathrm{ord}(\varphi_{i})\geq l$.

###### Proof.

Let $G^{(k,l)}\in K[t]\\{z_{1},\ldots,z_{n}\\}$ be the system obtained after
the transformation as in Section 2.1. Solutions
$(\varphi_{1},\ldots,\varphi_{n})\in K(\\!(t^{1/k})\\!)^{n}$ of $G$ are in
one-to-one correspondence to solutions $(\psi_{1},\ldots,\psi_{n})\in
K[\\![t]\\!]^{n}$ of $G^{(k,l)}$. By [DL84, Theorem 3.1], there is an
algorithm for deciding the solvability of $G^{(k,l)}$ in $K[\\![t]\\!]$. ∎

######  Example 4.2.

Let us consider $F=xx^{\prime}-1$ together with the initial condition
$x(0)=0$.111In Theorem 4.1, one can additionally impose a finite number of
equations and inequations for the Taylor coefficients of solutions as it can
also be seen in the proof of [DL84, Theorem 3.1]. The solutions are
$\varphi=\pm\sqrt{2t}\in\mathbb{Q}[\\![t^{1/2}]\\!]$, which are not formal
power series. Let us consider $F^{(2,1)}=zz^{\prime}-2t$ and $z(0)=0$. We see
that $\psi=\pm\sqrt{2}\cdot t\in\mathbb{Q}[\\![t]\\!]$. This solution is
easily found by [DL84, Theorem 3.1].

Other results on formal power series solutions might be generalized in a
similar way. For example, let $G$ be a given differential system and let
$G^{(k,l)}$ be the transformed system as in Section 2.1. Then results on jet
spaces (see e.g. [MS10]), constructed from $G^{(k,l)}$ and in one-to-one
correspondence to its formal power series solutions [Sei10, Remark 2.3.13]),
can be applied. In this way, a rigorous approximative description of the
formal Puiseux series solutions of $G$ can be given. Moreover, following
[Sei10, Section 9], sufficiently often differentiable solutions, whose
asymptotic behaviour are described by a Puiseux polynomial (they are called
geometric solutions in the reference), can be studied in this way allowing a
much bigger class of solutions under investigation.

### Acknowledgments

First author is partially supported by the grant PID2020-113192GB-I00
(Mathematical Visualization: Foundations, Algorithms and Applications) from
the Spanish MICINN and the OeAD project FR 09/2022.

## References

  * [ACJ03] Fuensanta Aroca, Josè Cano, and Françoise Jung.  Power series solutions for non-linear pde’s.  In Proceedings of the 2003 international symposium on Symbolic and algebraic computation, pages 15–22, 2003.
  * [AGT16] Fuensanta Aroca, Cristhian Garay, and Zeinab Toghani.  The fundamental theorem of tropical differential algebraic geometry.  Pacific Journal of Mathematics, 283(2):257–270, 2016.
  * [BFNS21] François Boulier, Sebastian Falkensteiner, Marc Paul Noordman, and Omar Leon Sanchez.  On the relationship between differential algebra and tropical differential algebraic geometry.  In Computer Algebra in Scientific Computing: 23rd International Workshop, CASC 2021, Sochi, Russia, September 13–17, 2021, Proceedings 23, pages 62–77. Springer, 2021.
  * [BH19] François Boulier and Mercedes Haiech.  The Ritt-Raudenbush Theorem and Tropical Differential Geometry.  working paper or preprint, 2019.
  * [DL84] Jan Denef and Leonard Lipshitz.  Power series solutions of algebraic differential equations.  Mathematische annalen, 267:213–238, 1984.
  * [Edg09] Gerald A. Edgar.  Transseries for Beginners.  Real Analysis Exchange, 35(2):253 – 310, 2009.
  * [FGLH+20] Sebastian Falkensteiner, Cristhian Garay-López, Mercedes Haiech, Marc Paul Noordman, Zeinab Toghani, and François Boulier.  The fundamental theorem of tropical partial differential algebraic geometry.  In Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, pages 178–185, 2020.
  * [FMSW23] Sebastian Falkensteiner, Johann Mitteramskogler, J Sendra, and Franz Winkler.  The algebro-geometric method: Solving algebraic differential equations by parametrizations.  Bulletin of the American Mathematical Society, 60(1):85–122, 2023\.
  * [GG16] Jeffrey Giansiracusa and Noah Giansiracusa.  Equations of tropical varieties.  Duke Math. J., 165(18):3379–3433, 2016.
  * [GM21] Jeffrey Giansiracusa and Stefano Mereta.  A general framework for tropical differential equations.  arXiv preprint arXiv:2111.03925, 2021.
  * [Gri17] Dima Grigoriev.  Tropical differential equations.  Advances in Applied Mathematics, 82:120–128, 2017.
  * [Kol73] Ellis Robert Kolchin.  Differential algebra & algebraic groups.  Academic press, 1973.
  * [LH14] Markus Lange-Hegermann.  Counting solutions of differential equations.  PhD thesis, Aachen, Techn. Hochsch., Diss., 2014.
  * [Mer22] Stefano Mereta.  A general framework for tropical differential equations.  PhD thesis, Swansea University and Université Grenoble Alpes, 2022.
  * [Mer23] Stefano Mereta.  The fundamental theorem of tropical differential algebra over nontrivially valued fields and the radius of convergence of nonarchimedean differential equations.  arXiv preprint arXiv:2303.12124, 2023.
  * [MR18] Diane Maclagan and Felipe Rincón.  Tropical ideals.  Compositio Mathematica, 154(3):640–670, 2018.
  * [MS10] Rahim Moosa and Thomas Scanlon.  Jet and prolongation spaces.  Journal of the Institute of Mathematics of Jussieu, 9(2):391–430, 2010.
  * [MS21] Diane Maclagan and Bernd Sturmfels.  Introduction to tropical geometry, volume 161.  American Mathematical Society, 2021.
  * [Rob14] Daniel Robertz.  Formal algorithmic elimination for PDEs, volume 2121.  Springer, 2014.
  * [Sei10] Werner M. Seiler.  Involution, volume 24.  Springer, 2010.
  * [vdH01] Joris van der Hoeven.  Complex transseries solutions to algebraic differential equations.  Université de Paris-Sud. Département de Mathématique, 2001\.
  * [VdH06] Joris Van der Hoeven.  Transseries and real differential algebra, volume 1888.  Springer, 2006.