Algebraic elements over the ring of power series

TL;DR

This paper establishes a precise criterion for when certain generalized power series are algebraic over power series rings with finite field coefficients, extending a classical theorem by Huang-Stefanescu.

Contribution

It provides a necessary and sufficient condition for algebraicity of generalized power series over finite fields, broadening the scope of existing algebraic criteria.

Findings

Derived a new algebraic criterion for generalized power series

Extended Huang-Stefanescu's classical theorem

Enhanced understanding of algebraic power series over finite fields

Abstract

We give a necessary and sufficient condition for a type of generalized power series to be algebraic over the ring of power series with coefficients in a finite field. This result extend a classical theorem of Huang-Stefanescu.

Equations

$$Z^p-Z-t^{-1}\in \mathbb{k}((t))[Z]$$

$$Z^p-Z-t^{-1}=\prod _{i=0}^{p-1}(Z-i-\sum _{j=1}^{\infty }{t}^{-\frac{1}{p^j}}).$$

$$f(t)=\sum _i{a}_i{t}^{\frac{s_i}{m{p}^{n_i}}}$$

$$f=\sum _{\alpha \in \Gamma }{c}_{\alpha }{t}^{\alpha }$$

$$\sum _{\alpha \in \Gamma }{c}_{\alpha }{t}^{\alpha }+\sum _{\alpha \in \Gamma }{d}_{\alpha }{t}^{\alpha }=\sum _{\alpha \in \Gamma }(c_{\alpha }+d_{\alpha })t^{\alpha }$$

$$(\sum _{{\alpha }^{\prime }\in \Gamma }{c}_{{\alpha }^{\prime }}{t}^{{\alpha }^{\prime }})(\sum _{{\alpha }^{\prime \prime }\in \Gamma }{d}_{{\alpha }^{\prime \prime }}{t}^{{\alpha }^{\prime \prime }})=\sum _{\alpha \in \Gamma }\sum _{{\alpha }^{\prime }+{\alpha }^{\prime \prime }=\alpha }{c}_{{\alpha }^{\prime }}{d}_{{\alpha }^{\prime \prime }}{t}^{\alpha }.$$

$$\begin{array}[]{r@{\hspace{1 pt}} c@{\hspace{1 pt}}c@{\hspace{4pt}}l}\hskip 1.0&\varphi_{1}f_{1}+\cdots+\varphi_{l}f_{l}=0,\hfil\hskip 1.0\end{array}$$

$$\begin{array}[]{r@{\hspace{1 pt}} c@{\hspace{1 pt}}c@{\hspace{4pt}}l}\hskip 1.0&0=\sum_{n,m}(\sum_{k=1}^{l}\varphi_{k,(n,m)}f_{k})t_{1}^{n}t_{2}^{m}.\hfil\hskip 1.0\end{array}$$

$$\mathrm{T}_{c}:=\bigg{\{}-\frac{b_{1}}{p}-\frac{b_{2}}{p^{2}}-\cdots\mid b_{i}\in\mathrm{A}_{p},\sum b_{i}\leq c\bigg{\}}.$$

$$a_{n+d}^{p^d}=-c_d^{-1}(c_0{a}_n+c_1{a}_{n+1}^p+\cdots +c_{d-1}{a}_{n+d-1}^{p^{d-1}})$$

$$\{(a_n,a_{n+1},...,a_{n+d-1})\mid n\in {\mathbb{Z}}_{>0}\}$$

$$a_n=f_{(-\frac{{\upsilon }_1}{p^{i_1+n}}-\cdots -\frac{{\upsilon }_e}{p^{i_e+n}},-\frac{{\omega }_1}{p^{j_1+n}}-\cdots -\frac{{\omega }_k}{p^{j_k+n}})}$$

$$\begin{array}[]{r@{\hspace{1 pt}} c@{\hspace{1 pt}}c@{\hspace{4pt}}l}d_{0}a_{n}^{p}+d_{1}a_{n+1}^{p^{2}}+\cdots+d_{RN-1}a_{n+RN-1}^{p^{RN}}=0\hskip 1.0\end{array}$$

$$d_0{f}_{p^{RN-1}v}^p+d_1{f}_{p^{RN-2}v}^{p^2}+\cdots +d_{RN-1}{f}_v^{p^{RN}}=0.$$

$$a_n=f_{(-\frac{{\upsilon }_1}{p^{i_1-RN+1-M+n}}-\cdots -\frac{{\upsilon }_e}{p^{i_e-RN+1-M+n}},-\frac{{\omega }_1}{p^{j_1-RN+1-M+n}}-\cdots -\frac{{\omega }_k}{p^{j_k-RN+1-M+n}})}.$$

$$d_0{f}_{p^{RN-1}v}^p+d_1{f}_{p^{RN-2}v}^{p^2}+\cdots +d_{RN-1}{f}_v^{p^{RN}}=d_0{a}_M^p+d_1{a}_{M+1}^{p^2}+\cdots +d_{RN-1}{a}_{M+RN-1}^{p^{RN}}=0,$$

$$f=\sum _i{f}_{(i,-\frac{{\omega }_1}{p^{j_1}}-\cdots -\frac{{\omega }_k}{p^{j_k}})}{t}_1^i{t}_2^{-\frac{{\omega }_1}{p^{j_1}}-\cdots -\frac{{\omega }_k}{p^{j_k}}}$$

$$a_n=f_{(-\frac{{\upsilon }_1}{p^{i_1}}-\cdots -\frac{{\upsilon }_{l-1}}{p^{i_{l-1}}}-\frac{1}{p^n}(\frac{{\upsilon }_l}{p^{i_l}}+\cdots +\frac{{\upsilon }_e}{p^{i_e}}),-\frac{{\omega }_1}{p^{j_1}}-\cdots -\frac{{\omega }_k}{p^{j_k}})}$$

$$\begin{array}[]{r@{\hspace{1 pt}} c@{\hspace{1 pt}}c@{\hspace{4pt}}l}\hskip 1.0&\varphi_{0}f^{\prime}+\varphi_{1}f^{\prime p}+\cdots+\varphi_{l}f^{\prime p^{l}}=0,\hfil\hskip 1.0\end{array}$$

$$m:=\min \{{\alpha }_2\mid \exists {\alpha }_1\mathrm{with}({\alpha }_1,{\alpha }_2)\in \mathrm{supp}({\phi }_k)\mathrm{for}\mathrm{some}k\in \{0,...,l\}\}.$$

$$\begin{array}[]{r@{\hspace{1 pt}} c@{\hspace{1 pt}}c@{\hspace{4pt}}l}\hskip 1.0&\varphi_{0}^{\prime}f^{\prime}+\varphi_{1}^{\prime}f^{\prime p}+\cdots+\varphi_{l}^{\prime}f^{\prime p^{l}}=0,\hfil\hskip 1.0\end{array}$$

$${\phi }_k^{\prime }={\phi }_k^{(1)}+{\phi }_k^{(2)}$$

$$\begin{array}[]{r@{\hspace{1 pt}} c@{\hspace{1 pt}}c@{\hspace{4pt}}l}\hskip 1.0&\varphi_{0}^{(1)}f^{\prime}+\cdots+\varphi_{l}^{(1)}f^{\prime p^{l}}=-\varphi_{0}^{(2)}f^{\prime}-\cdots-\varphi_{l}^{(2)}f^{\prime p^{l}}\hfil\hskip 1.0\end{array}$$

$$f=\sum _j{f}_{(-\frac{{\upsilon }_1}{p^{i_1}}-\cdots -\frac{{\upsilon }_e}{p^{i_e}},j)}{t}_1^{-\frac{{\upsilon }_1}{p^{i_1}}-\cdots -\frac{{\upsilon }_e}{p^{i_e}}}{t}_2^j$$

$$a_n=f_{(-\frac{{\upsilon }_1}{p^{i_1}}-\cdots -\frac{{\upsilon }_{l-1}}{p^{i_{l-1}}}-\frac{1}{p^n}(\frac{{\upsilon }_l}{p^{i_l}}+\cdots +\frac{{\upsilon }_e}{p^{i_e}}),-\frac{{\omega }_1}{p^{j_1}}-\cdots -\frac{{\omega }_{r-1}}{p^{j_{r-1}}}-\frac{1}{p^n}(\frac{{\omega }_r}{p^{j_r}}+\cdots +\frac{{\omega }_k}{p^{j_k}}))}$$

$$b_n=f_{(-\frac{{\upsilon }_1}{p^{i_1}}-\cdots -\frac{{\upsilon }_e}{p^{i_e}},-\frac{{\omega }_1}{p^{j_1}}-\cdots -\frac{{\omega }_{r-1}}{p^{j_{r-1}}}-\frac{1}{p^n}(\frac{{\omega }_r}{p^{j_r}}+\cdots +\frac{{\omega }_k}{p^{j_k}}))},$$

$$c_n=f_{(-\frac{{\upsilon }_1}{p^{i_1}}-\cdots -\frac{{\upsilon }_{l-1}}{p^{i_{l-1}}}-\frac{1}{p^n}(\frac{{\upsilon }_l}{p^{i_l}}+\cdots +\frac{{\upsilon }_e}{p^{i_e}}),-\frac{{\omega }_1}{p^{j_1}}-\cdots -\frac{{\omega }_k}{p^{j_k}})}$$

$$\begin{array}[]{r@{\hspace{1 pt}} c@{\hspace{1 pt}}c@{\hspace{4pt}}l}d_{0}a_{n}^{p}+d_{1}a_{n+1}^{p^{2}}+\cdots+d_{2rN-1}a_{n+2rN-1}^{p^{2rN}}=0\hskip 1.0\end{array}$$

$$g:=d_0{f}^{1/p^{2rN-1}}+d_1{f}^{1/p^{2rN-2}}+\cdots +d_{2rN-1}f.$$

$$g_{j^{\prime }}=d_0{f}_{p^{2rN-1}{j}^{\prime }}^{1/p^{2rN-1}}+d_1{f}_{p^{2rN-2}{j}^{\prime }}^{1/p^{2rN-2}}+\cdots +d_{2rN-1}{f}_{j^{\prime }},$$

Full text

1. Introduction

This work is concerning to the problem of describing algebraic elements over the ring of power series. It is well known that when $\Bbbk$ is an algebraically closed field of characteristic zero, the algebraic closure of the field of power series $\Bbbk((t))$ is the so-called field of Puiseux series. That is, the field formed by the union of all the fields $\Bbbk((t^{1/d}))$ where $d$ is a positive integer. This result is known as the Newton-Puiseux theorem (see for example [2] for a formal presentation). When $\Bbbk$ is an algebraically closed field of characteristic $p>0,$ Chevalley [3] noted that the polynomial

[TABLE]

has no root in the field of Puiseux series. In fact, the Abhyankar*,*s paper [1] shows that this polynomial can be factored as follows

[TABLE]

Using this factorization Huang [4] considered generalized power series of the form

[TABLE]

where $m\in\mathbb{Z}_{>0},$ $n_{i}\in\mathbb{Z}_{\geq 0}$ and $s_{i}\in\mathbb{Z}$ and the support of $f$ is a well-ordered set. The set of all generalized power series of this type is a field and Huang [4] proved that contains an algebraic closure of the field $\Bbbk((t)).$ If $\Bbbk$ is a perfect field of positive characteristic, the algebraic closure of $\Bbbk((t))$ consist of the field of the so-called twist-recurrent series, and it is a result of Kedlaya [6] (see also [7]). The twist-recurrent series are generalized power series that hold two technical conditions, one over the exponents of the series and another one over the coefficients (see definition in [6]). As example of this type of series are the series that appear in the following theorem given independently by Huang [4] and Stef$\mathrm{\breve{a}}$nescu [11] (see also [12]).

Theorem 1.1**.**

(Huang, Stef$\breve{a}$nescu) The series $f(t)=\sum_{i=1}^{\infty}a_{i}t^{\frac{-1}{p^{i}}}\in\overline{\mathbb{F}_{p}}((t^{\mathbb{Q},p}))$ is algebraic over $\overline{\mathbb{F}_{p}}((t))$ if and only if the sequence $\{a_{i}\}$ is eventually periodic.

Theorem 1.1 can be also deduced from the main result of [6]. S. Vaidya [13] extend the criterion of Huang-Stef$\mathrm{\breve{a}}$nescu to a certain type of functions, in the case that $\Bbbk$ is not equal to algebraic closure of its prime field.

In this paper we present the analogue to Theorem 1.1 for generalized power series in two variables with coefficients in a finite field (Corollary 3.7).

2. Generalized power series

Let $\Gamma$ be a totally ordered group and let $\Bbbk$ be a field. The field of Hahn series $\Bbbk((t^{\Gamma}))$ is defined to be the collection of all elements of the form

[TABLE]

with $c_{\alpha}\in\Bbbk$ such that the set of exponents of $f$ is a well-ordered set. The sum and product are given by

[TABLE]

and

[TABLE]

The support of a series $f$ is the set $\{\alpha\in\Gamma\mid c_{\alpha}\neq 0\}.$ The field $\Bbbk((t^{\Gamma}))$ is also called the field of **generalized power series **over $\Bbbk$ with support in $\Gamma.$ It is known that when $\Bbbk$ is algebraically closed and $\Gamma$ is a divisible group, the field $\Bbbk((t^{\Gamma}))$ is algebraically closed [5].

Rayner noted that we can take the set of series with support in a proper subfamily of the family of all well-ordered subsets of $\Gamma$ and still lead a field. He called this subfamily a field-family (see definition in [8]). With the notion of field-family, a family of of algebraically closed fields of series containing the ring of power series in several variables is given in [10, Theorem 5.3].

Let us denote by $\Bbbk((t_{1}^{\mathbb{Q}},t_{2}^{\mathbb{Q}})),$ the field of generalized power series with coefficients in $\Bbbk$ and support a well-ordered subset of $\mathbb{Q}\times\mathbb{Q}.$

Lemma 2.1**.**

Suppose that $f_{1},...,f_{l}\in\Bbbk((t_{1}^{\mathbb{Q}},t_{2}^{\mathbb{Q}}))$ with support in $(-1,0]\times(-1,0]$ and that $f_{1},...,f_{l}$ are linearly dependent over $\Bbbk((t_{1},t_{2})).$ Then $f_{1},...,f_{l}$ are also linearly dependent over $\Bbbk.$

Proof.

Since $f_{1},...,f_{l}$ are linearly dependent over $\Bbbk((t_{1},t_{2})),$ we can obtain a nonzero linear relation of the form,

[TABLE]

where $\varphi_{k}\in\Bbbk[[t_{1},t_{2}]]$ $\forall k=1,...,l.$ We can write $\varphi_{k}=\sum_{n,m}\varphi_{k,(n,m)}t_{1}^{n}t_{2}^{m}$ for some $\varphi_{k,(n,m)}\in\Bbbk.$ Therefore

[TABLE]

Now note that the support of $(\sum_{k=1}^{l}\varphi_{k,(n,m)}f_{k})t_{1}^{n}t_{2}^{m}$ is contained in $(n-1,n]\times(m-1,m]$ and thus these supports are disjoint for different $n$ and $m.$ This implies that the summand in (2.2) must be zero for each $(n,m)$ and thus $\sum_{k=1}^{l}\varphi_{k,(n,m)}f_{k}=0$ for each $(n,m).$ Note that the $\varphi_{k,(n,m)}$ cannot all be zero because $\varphi_{1},..,\varphi_{l}$ would have all been zero. It follows that $f_{1},...,f_{l}$ are linearly dependent over $\Bbbk.$ ∎

From now on $\mathbb{F}$ will be a finite field.

Denote $\mathrm{A}_{p}:=\{0,1,...,p-1\}$ and for $c$ a nonnegative integer, let $\mathrm{T}_{c}$ be the subset given by

[TABLE]

We recall that a sequence $a_{n}$ is eventually periodic if there exist $s$ and $m$ such that $a_{n+s}=a_{n}$ for all $n\geq m$.

From the proof of Lemma 2.6 in [11], we can extract the following lemma.

Lemma 2.2**.**

Consider a sequence $\{a_{n}\}\subset\mathbb{F}.$ Suppose that there is $d,$ and $c_{0},...,c_{d}$ not all zero, such that $c_{0}a_{n}+c_{1}a_{n+1}^{p}+\cdots+c_{d}a_{n+d}^{p^{d}}=0,$ $\forall n\geq k_{0}$ for some $k_{0}\in\mathbb{Z}_{>0}.$ Then $\{a_{n}\}$ becomes eventually periodic.

Proof.

We may suppose that $c_{d}\neq 0.$ Dividing by the constant $c_{d}$ we get

[TABLE]

Thus for every $n\in\mathbb{Z}_{>0},$ we have that $a_{n+d}$ is completely determined by the $d-$tuple $(a_{n},a_{n+1},...,a_{n+d-1}).$ Since $a_{n}\in\mathbb{F}\,$ $\forall n$ and $\mathbb{F}$ is finite, we get that the set

[TABLE]

is finite. Therefore there are $r,t\in\mathbb{Z}_{>0},$ $r\neq t$ such that $(a_{r},a_{r+1},...,a_{r+d-1})=(a_{t},a_{t+1},...,a_{t+d-1}).$ This implies that $a_{r+d}^{p^{d}}=a_{t+d}^{p^{d}}$ and then $a_{r+d}=a_{t+d}.$ But this implies that $(a_{r+1},...,a_{r+d})=(a_{t+1},...,a_{t+d})$ and then $a_{r+d+1}=a_{t+d+1}.$ In general it follows that $a_{r+d+k}=a_{t+d+k}$ for every $k\in\mathbb{Z}_{\geq 0}.$ We may suppose that $r<t$ and let $s:=t-r.$ Then $a_{n+s}=a_{n+t-r}=a_{d+t+n-r-d}.$ Therefore if $n\geq r+d,$ we get that $a_{d+t+n-r-d}=a_{d+r+n-r-d}=a_{n}.$ That is, $a_{n+s}=a_{n}$ for every $n\geq m:=\mathrm{max}(r+d,k_{0}).$ ∎

Lemma 2.3**.**

Let $f=\sum_{i,j}f_{(i,j)}t_{1}^{i}t_{2}^{j}\in\mathbb{F}((t_{1}^{\mathbb{Q}},t_{2}^{\mathbb{Q}}))$ be a series with support in $\mathrm{T}_{c}\times\mathrm{T}_{c}.$ Suppose that there exist positive integers $M,N$ and $R$ such that there are $d_{0},...,d_{RN-1}\in\mathbb{F}$ not all zero such that every sequence $\{a_{n}\}_{n=0}^{\infty}$ of the form,

[TABLE]

satisfies

[TABLE]

for all $n\geq M.$ If $v$ and $p^{RN-1+M}v$ are in $\mathrm{T}_{c}\times\mathrm{T}_{c},$ then

[TABLE]

Proof.

Let say that $v:=(-\frac{\upsilon_{1}}{p^{i_{1}}}-\frac{\upsilon_{2}}{p^{i_{2}}}-\cdots-\frac{\upsilon_{e}}{p^{i_{e}}},\quad-\frac{\omega_{1}}{p^{j_{1}}}-\frac{\omega_{2}}{p^{j_{2}}}-\cdots-\frac{\omega_{k}}{p^{j_{k}}})$, therefore

$p^{RN-1+M}v=(-\frac{\upsilon_{1}}{p^{i_{1}-RN+1-M}}-\cdots-\frac{\upsilon_{e}}{p^{i_{e}-RN+1-M}},\quad-\frac{\omega_{1}}{p^{j_{1}-RN+1-M}}-\cdots-\frac{\omega_{k}}{p^{j_{k}-RN+1-M}}).$

Consider the sequence

[TABLE]

We get that

[TABLE]

by hypothesis.

∎

3. Algebraic series

Lemma 3.1**.**

Let $-\frac{\omega_{1}}{p^{j_{1}}}-\cdots-\frac{\omega_{k}}{p^{j_{k}}}$ be an element of $\mathrm{T}_{c}.$ Let

[TABLE]

be a series with support in $\mathrm{T}_{c}\times\mathrm{T}_{c}.$ Then $f$ is algebraic over $\mathbb{F}((t_{1},t_{2})),$ if and only if every sequence of the form $a_{n}=f_{(-\frac{\upsilon_{1}}{p^{i_{1}}}-\cdots-\frac{\upsilon_{l-1}}{p^{i_{l-1}}}-\frac{1}{p^{n}}(\frac{\upsilon_{l}}{p^{i_{l}}}+\cdots+\frac{\upsilon_{e}}{p^{i_{e}}}),\quad-\frac{\omega_{1}}{p^{j_{1}}}-\cdots-\frac{\omega_{k}}{p^{j_{k}}})}$ is eventually periodic.

Proof.

Suppose that every sequence of the form

[TABLE]

is eventually periodic. Is enough to see that $f^{\prime}:=\sum f_{(i,-\frac{\omega_{1}}{p^{j_{1}}}-\cdots-\frac{\omega_{k}}{p^{j_{k}}})}t_{1}^{i}$ is algebraic over $\mathbb{F}((t_{1},t_{2})).$ Note that $f^{\prime}\in\mathbb{F}((t_{1}^{\mathbb{Q},p})),$ so by [6, Theorem 15] $f^{\prime}$ is algebraic over $\mathbb{F}((t_{1}))$ and then over $\mathbb{F}((t_{1},t_{2})).$

Now suppose that $f$ is algebraic over $\mathbb{F}((t_{1},t_{2})).$ Then $f^{\prime}$ is algebraic over $\mathbb{F}((t_{1},t_{2})).$ So we can write,

[TABLE]

for some $l\in\mathbb{Z}_{\geq 0}$ and for some $\varphi_{k}\in\mathbb{F}[[t_{1},t_{2}]],$ $\forall k=0,...,l.$ Denote

[TABLE]

Multiply by $t_{2}^{-m}$ both sides of (3.1), we get

[TABLE]

where some of the $\varphi_{k}^{\prime}$ have some terms just depending on $t_{1},$ that is, terms with support of the form $\{(\alpha_{1},0)\}.$ For $k=0,...,l,$ we can write

[TABLE]

where $\varphi_{k}^{(1)}$ just contains the terms of $\varphi_{k}^{\prime}$ with support of the form $\{(\alpha_{1},0)\}$ and $\varphi_{k}^{(2)}$ contains the remaining terms. By (3.2), we can write,

[TABLE]

This equality implies that $\varphi_{0}^{(1)}f^{\prime}+\varphi_{1}^{(1)}f^{\prime p}+\cdots+\varphi_{l}^{(1)}f^{\prime p^{l}}=0,$ because the terms in the left side of (3.3) just depend on $t_{1}.$ This means that $f^{\prime}$ is algebraic over $\mathbb{F}[[t_{1}]]$ then again by [6, Theorem 15], we get that every sequence of the form $a_{n}$ is eventually periodic.

∎

Lemma 3.2**.**

Let $-\frac{\upsilon_{1}}{p^{i_{1}}}-\cdots-\frac{\upsilon_{e}}{p^{i_{e}}}$ be an element of $\mathrm{T}_{c}.$ Let

[TABLE]

be a series with support in $\mathrm{T}_{c}\times\mathrm{T}_{c}.$ Then $f$ is algebraic over $\mathbb{F}((t_{1},t_{2})),$ if and only if every sequence of the form $a_{n}=f_{(-\frac{\upsilon_{1}}{p^{i_{1}}}-\cdots-\frac{\upsilon_{e}}{p^{i_{e}}},\quad-\frac{\omega_{1}}{p^{j_{1}}}-\cdots-\frac{\omega_{r-1}}{p^{j_{r-1}}}-\frac{1}{p^{n}}(\frac{\omega_{r}}{p^{j_{r}}}+\cdots+\frac{\omega_{k}}{p^{j_{k}}}))}$ is eventually periodic.

Proof.

Apply a similar argument as in the proof of Lemma 3.1. ∎

Theorem 3.3**.**

Let $f=\sum_{i,j}f_{i,j}t_{1}^{i}t_{2}^{j}\in\mathbb{F}((t_{1}^{\mathbb{Q}},t_{2}^{\mathbb{Q}}))$ be a series with support in $\mathrm{T}_{c}\times\mathrm{T}_{c}.$ Suppose that there exist positive integers $M$ and $N$ such that every sequence $\{a_{n}\}_{n=0}^{\infty}$ of the form,

[TABLE]

has period $N$ after $M$ terms and the sequences of the form

[TABLE]

[TABLE]

are eventually periodic. Then $f$ is algebraic over $\mathbb{F}((t_{1},t_{2})).$

Proof.

Let $p^{r}$ be the cardinality of $\mathbb{F},$ where $r\in\mathbb{Z}_{\geq 0}.$ There exist $d_{0},...,d_{2rN-1}\in\mathbb{F}$ not all zero such that every sequence $a_{n}$ satisfies

[TABLE]

for all $n\geq M.$

Indeed, note that

$a_{n}^{p}+a_{n+1}^{p^{2}}+\cdots+a_{n+rN-1}^{p^{rN}}-a_{n+rN}^{p^{rN+1}}-\cdots-a_{n+2rN-1}^{p^{2rN}}=0.$ Thus $d_{s}=1$ for $s=0,...,rN-1$ and $d_{s}=-1$ for $s=rN,rN+1,...,2rN-1,$ is a solution.

Consider one of theses solutions $(d_{0},...,d_{2rN-1})$ and consider the series

[TABLE]

If $g=0,$ $f$ is algebraic. So we can assume that $g\neq 0.$ We are going to show that $g$ is a finite sum of algebraic series, from which it follows that $g^{p^{2rN}}$ is algebraic and thus $f.$

Note that the coefficient $g_{j^{\prime}}$ of $g$ is

[TABLE]

where $j^{\prime}\in\mathrm{supp}(f).$

Take

[TABLE]

With out loss of generality suppose that $e\leq k.$ Note that there is $m_{0}\in\mathbb{Z}_{>0}$ such that $\frac{b}{p^{n-2rN+1-M}}<\frac{1}{k}$ $\forall n\geq m_{0}$ and for any $b\in\{0,1,2,...,p-1\}.$

If $j^{\prime}$ holds that $i_{1}^{\prime},...,i_{e}^{\prime}\geq m_{0}$ and $j_{1}^{\prime},...,j_{k}^{\prime}\geq m_{0}$ then $p^{2rN-1+M}j^{\prime}\in\mathrm{T}_{c}\times\mathrm{T}_{c}$ because

[TABLE]

[TABLE]

Thus by Lemma 2.3 we can conclude that $g_{j^{\prime}}^{p^{2rN}}=0$ and then $g_{j^{\prime}}=0.$ We can sort out the remaining terms of $g$ whose supports use $\{\upsilon_{1},...,\upsilon_{e}\}$ in the first coordinate and $\{\omega_{1},...,\omega_{k}\}$ in the second coordinate, in series with support of the following forms

i)

Series with support of the form $(-\frac{\upsilon_{1}}{p^{i_{1}}}-\cdots-\frac{\upsilon_{e}}{p^{i_{e}}},\quad-\frac{\omega_{1}}{p^{j_{1}}}-\cdots-\frac{\omega_{k}}{p^{j_{k}}}),$ where some of the indices $i_{s}$ for $s\in\{1,...,e\}$ are variable indices (these variable indices are in $\mathbb{Z}_{>0}$) and all the remaining indices are constant and these constant indices satisfy that belongs to the set $\{1,...,m_{0}-1\}.$

ii)

Series with support of the form $(-\frac{\upsilon_{1}}{p^{i_{1}}}-\cdots-\frac{\upsilon_{e}}{p^{i_{e}}},\quad-\frac{\omega_{1}}{p^{j_{1}}}-\cdots-\frac{\omega_{k}}{p^{j_{k}}}),$ where some of the $j_{t}$ for $t\in\{1,...,k\}$ are variable indices and all the remaining indices are constant and they satisfy that belongs to the set $\{1,...,m_{0}-1\}.$

iii)

Series with support of the form $(-\frac{\upsilon_{1}}{p^{i_{1}}}-\cdots-\frac{\upsilon_{e}}{p^{i_{e}}},\quad-\frac{\omega_{1}}{p^{j_{1}}}-\cdots-\frac{\omega_{k}}{p^{j_{k}}})$ where some of the indices $i_{s}$ for $s\in\{1,...,e\}$ are constant and they satisfy that $i_{s}\in\{1,...,m_{0}-1\}$ and where some of the $j_{t}$ for $t\in\{1,...,k\}$ are constant and they satisfy that $j_{t}\in\{1,...,m_{0}-1\}$ and the remaining indices are variable indices.

Note that there are many finitely series with support as $\mathrm{\textbf{i}}),\mathrm{\textbf{ii}})$ and $\mathrm{\textbf{iii}}).$ We are going to show that these series are algebraic.

Take one of this series, let say $f^{(1)}.$ Suppose that the series $f^{(1)}$ has some of the indices $i_{s}$ as variable indices and some of the $j_{t}$ also as variable indices. Without loss of generality suppose that $i_{1}^{\prime},...,i_{l}^{\prime}$ and $j_{1}^{\prime},...,j_{m}^{\prime}$ are the constant indices. Note that it is enough to show that the following series is algebraic,

[TABLE]

where the sum is running over all $i_{l+1},...,i_{e},j_{m+1},...,j_{k}\in\mathbb{Z}_{>0}$ such that the $2$-tuple

[TABLE]

We denote this series again by $f^{(1)}.$ Note that sequences of the form

$e_{n}=g_{(-\frac{\upsilon_{1}}{p^{i_{1}^{\prime}}}-\cdots-\frac{\upsilon_{l}}{p^{i_{l}^{\prime}}}-\frac{1}{p^{n}}(-\frac{\upsilon_{l+1}}{p^{i_{l+1}^{\prime}}}-\cdots-\frac{\upsilon_{e}}{p^{i_{e}^{\prime}}}),\;-\frac{\omega_{1}}{p^{j_{1}^{\prime}}}-\cdots-\frac{\omega_{m}}{p^{j_{m}^{\prime}}}-\frac{1}{p^{n}}(-\frac{\omega_{m+1}}{p^{j_{m+1}^{\prime}}}-\cdots-\frac{\omega_{k}}{p^{j_{k}^{\prime}}}))}$ are eventually periodic because $e_{n}$ is a sum of sequences of period $N$ after $M$ terms, explicitly we have

[TABLE]

[TABLE]

[TABLE]

where $\ast$ is $-\frac{\omega_{1}}{p^{j_{1}^{\prime}-(2rN-1)}}-\cdots-\frac{\omega_{m}}{p^{j_{m}^{\prime}-(2rN-1)}}-\frac{1}{p^{n-(2rN-1)}}(-\frac{\omega_{m+1}}{p^{j_{m+1}^{\prime}}}-\cdots-\frac{\omega_{k}}{p^{j_{k}^{\prime}}})$

At this point, we can repeat all the above argument for $f^{(1)}$ instead of $f$ and $e_{n}$ instead of $a_{n}.$ That is, we can find $d_{0}^{(1)},...,d_{2rN-1}^{(1)}\in\mathbb{F}$ not all zero such that

[TABLE]

for all $n\geq M.$ Consider the series

[TABLE]

If $g^{(1)}=0$ then $f^{(1)}$ is algebraic. So we can assume that $g^{(1)}\neq 0.$

We are going to show that $g^{(1)}$ is algebraic, from this it will follows that $f^{(1)}$ is algebraic. Reasoning analogously as before we can see now that $g^{(1)}$ is a finite sum of series such that the support of these series hold that at least one of the following is true: the number of variable indices $i_{s}$ is less than $e-l$ or the number of variable indices $j_{t}$ is less than $k-m.$ Take one of this series, let say $f^{(2)},$

[TABLE]

where $a>1$ or $b>1,$ and the sum is running over all $i_{l+a},...,i_{e},j_{m+b},...,j_{k}\in\mathbb{Z}_{>0}$ and

[TABLE]

Reasoning as before we have that sequences of the form

$e_{n}^{\prime}=g_{(-\frac{\upsilon_{1}}{p^{i_{1}^{\prime}}}-\cdots-\frac{\upsilon_{l+a-1}}{p^{i_{l+a-1}^{\prime}}}-\frac{1}{p^{n}}(\frac{\upsilon_{1+a}}{p^{i_{l+a}^{\prime}}}+\cdots+\frac{\upsilon_{e}}{p^{i_{e}^{\prime}}}),\;-\frac{\omega_{1}}{p^{j_{1}^{\prime}}}-\cdots-\frac{\omega_{m+b-1}}{p^{j_{m+b-1}^{\prime}}}-\frac{1}{p^{n}}(\frac{\omega_{m+b}}{p^{j_{m+b}^{\prime}}}-\cdots-\frac{\omega_{k}}{p^{j_{k}^{\prime}}}))}$ are eventually periodic. Now we can repeat the above argument recursively for $f^{(2)},$ we will show that

[TABLE]

is algebraic. In general, we get

[TABLE]

where $g^{(n)}$ is a finite sum of series and the support of these series holds that at least one of the following is true: the number of variable indices $i_{s}$ is less than the number of variable indices of the exponents of $f^{(n)}$ in the first coordinate or the number of variable indices $j_{t}$ is less than the number of variable indices of the exponents of $f^{(n)}$ in the second coordinate. So eventually, in many finitely steps we will get sequences of the following forms

[TABLE]

where the sum is running over all $i_{e},j_{k}\in\mathbb{Z}_{>0},$ and

[TABLE]

because on each step the number of variable indices is strictly decreasing. So as before we can get a series

[TABLE]

The series $g^{(n_{0})}$ is a sum of a finite number of series of the following forms

[TABLE]

where $i$ is a constant, or

[TABLE]

where $j$ is a constant. We can apply the hypothesis that sequences as $(b_{n})$ and $(c_{n})$ are eventually periodic and Lemma 3.1 or Lemma 3.2 to conclude that these series are algebraic over $\mathbb{F}((t_{1},t_{2})).$ It follows that series with support as $\mathrm{\textbf{i}}),\mathrm{\textbf{ii}})$ and $\mathrm{\textbf{iii}})$ are algebraic over $\mathbb{F}((t_{1},t_{2})).$ Since there are many finitely elections of the $\upsilon_{i}^{,}$s and $\omega_{i}^{,}$s such that $\sum\upsilon_{i}\leq c$ and $\sum\omega_{j}\leq c,$ it follows that $g$ is a finite sum of algebraic series. Thus $f$ is algebraic over $\mathbb{F}((t_{1},t_{2})).$ ∎

Let $A$ be a ring. Let $\Gamma$ be an abelian totally ordered group and let $\infty$ be an element such that $x<\infty$ for every $x$ in $\Gamma$. Extend the law on $\Gamma\cup\{\infty\}$ by $\infty+x=\infty+\infty=\infty.$ A map

[TABLE]

is said to be a valuation of $A$ if satisfies the following properties for all $x,y\in A:$

i)

$\nu(x\cdot y)=\nu(x)+\nu(y),$

ii)

$\nu(x+y)\geq\mathrm{min}(\nu(x),\nu(y)).$

iii)

$\nu(x)=\infty$ if and only if $x=0.$

Remark 3.4*.*

[14, Remark 1.3] If we have a mapping $\nu:A\longrightarrow\Gamma\cup\{\infty\}$ with conditions $\mathrm{\textbf{i}}),\mathrm{\textbf{ii}})$ and with $\nu(0)=\infty,$ but if we do$\mathrm{n^{,}}$t assume that $\nu$ takes the value $\infty$ only for $0,$ the set $\mathcal{P}=\nu^{-1}(\infty)$ is an ideal prime of $A$ and $\nu$ induces a valuation on the integral domain $A/\mathcal{P}.$

Proposition 3.5**.**

The second property of a valuation $\nu$ can be generalized for any set $\{x_{1},...,x_{n}\}$ in $A$ by $\nu(\sum_{i=1}^{n}x_{i})\geq\mathrm{min}(\nu(x_{1}),...,\nu(x_{n})).$ If the minimum is reached by only one of the $\nu(x_{i})$ we get the equality: $\nu(\sum_{i=1}^{n}x_{i})=\mathrm{min}(\nu(x_{1}),...,\nu(x_{n}))$.

Proof.

See for example [14, Proposition 1.3] ∎

Theorem 3.6**.**

Suppose that $\upsilon_{1},...,\upsilon_{a},$ and $\omega_{1},...,\omega_{b}$ belong to $\sum_{p}.$ Let

[TABLE]

be a series with support in $T_{{}_{\sum_{i=1}^{a}\nu_{i}}}\times T_{{}_{\sum_{j=1}^{b}\omega_{j}}}$ and suppose that $f$ is algebraic over $\mathbb{F}((t_{1},t_{2})).$ Then sequences of the form,

[TABLE]

[TABLE]

and

[TABLE]

are eventually periodic.

Proof.

Consider a sequence

[TABLE]

Since $f$ is algebraic over $\mathbb{F}((t_{1},t_{2})),$ so is

[TABLE]

Since $g$ is algebraic over $\mathbb{F}((t_{1},t_{2})),$ the extension $\mathbb{F}((t_{1},t_{2}))(g)/\mathbb{F}((t_{1},t_{2}))$ is finite. So we can choose any linear dependence among $g,g^{p},g^{p^{2}},...,g^{p^{m}},...,.$ Thus there are $d\in\mathbb{Z}_{>0}$ and $\varphi_{0},\varphi_{1},...,\varphi_{d}\in\mathbb{F}((t_{1},t_{2}))$ not all zero such that

[TABLE]

By clearing denominators, we may suppose that $\varphi_{0},\varphi_{1},...,\varphi_{d}\in\mathbb{F}[[t_{1},t_{2}]]$.

Now consider the following subseries of $g:$

[TABLE]

Let $h:=g-\overline{g}.$ We have,

[TABLE]

Let $\psi:=\varphi_{0}\overline{g}+\varphi_{1}\overline{g}^{p}+\cdots+\varphi_{d}\overline{g}^{p^{d}}$ and $\psi^{\prime}:=\varphi_{0}h+\varphi_{1}h^{p}+\cdots+\varphi_{d}h^{p^{d}}.$

The exponents of the terms from $\varphi_{t}\overline{g}^{p^{t}}$ for $t\in\{0,1,...,d\}$ are of the form

[TABLE]

Fix $t\in\{0,1,...,d\}.$ Note that there is $k_{t}$ such that

$0<\frac{\upsilon_{l}}{p^{i_{l}^{\prime}+n-t}}+\cdots+\frac{\upsilon_{a}}{p^{i_{a}^{\prime}+n-t}}<1$ and $0<\frac{\omega_{r}}{p^{j_{r}^{\prime}+n-t}}+\cdots+\frac{\omega_{b}}{p^{j_{b}^{\prime}+n-t}}<1,$ for all $n\geq k_{t}.$

We are going to show that the terms from $\varphi_{t}\overline{g}^{p^{t}}$ whose exponents satisfy in their expressions (3.7) that $n\geq k_{t},$ cannot cancel with the terms from $\varphi_{t}h^{p^{t}}.$

Consider a point in the support of $\varphi_{t}h^{p^{t}}.$ The first coordinate of this point is of the form

[TABLE]

and the second coordinate is of the form

[TABLE]

We will refer to this point as the point $\mathbf{A}.$

Here the $i_{1}^{\prime},...,i_{l-1}^{\prime},i_{1},...,i_{a}$ and $j_{1}^{\prime},...,j_{r-1}^{\prime},j_{1},...,j_{b}$ that appears in $(3.5)$ satisfy that

[TABLE]

Fix a point as (3.7) with $n\geq k_{t}$ (we will refer to this point as the point $\mathbf{B}$) and the point $\mathbf{A}.$ Suppose that these points are equal.

i)

Suppose that $(3.5)$ holds that $i_{1}^{\prime},...,i_{l-1}^{\prime},i_{1},\ldots,i_{a}>t$ and $j_{1}^{\prime},...,j_{r-1}^{\prime},j_{1},...,j_{b}>t.$

• •

Suppose that at least one of the following is true: not any numbers $\frac{\upsilon_{1}}{p^{i_{1}^{\prime}-t}},\cdots,\frac{\upsilon_{l-1}}{p^{i_{l-1}^{\prime}-t}}$ at all are sumands in $\frac{\upsilon_{1}}{p^{i_{1}-t}}+\cdots+\frac{\upsilon_{a}}{p^{i_{a}-t}}$, or not any numbers $\frac{\omega_{1}}{p^{j_{1}^{\prime}-t}},\cdots,\frac{\omega_{r-1}}{p^{j_{r-1}^{\prime}-t}}$ at all are sumands in $\frac{\omega_{1}}{p^{j_{1}-t}}+\cdots+\frac{\omega_{b}}{p^{j_{b}-t}}.$

Without loss of generality suppose that we are in the first case. Note that

[TABLE]

and

[TABLE]

Clearly if

[TABLE]

or

[TABLE]

the points $\mathbf{A}$ and $\mathbf{B}$ cannot be equal.

If $\frac{\upsilon_{1}}{p^{i_{1}^{\prime}-t}}+\cdots+\frac{\upsilon_{l-1}}{p^{i_{l-1}^{\prime}-t}}-\frac{\upsilon_{1}}{p^{i_{1}-t}}-\cdots-\frac{\upsilon_{a}}{p^{i_{a}-t}}<0,$ the equality of the first coordinate of $\mathbf{A}$ and $\mathbf{B}$ implies that

[TABLE]

and then

[TABLE]

Thus we obtain a contradiction.

If $\frac{\upsilon_{1}}{p^{i_{1}^{\prime}-t}}+\cdots+\frac{\upsilon_{l-1}}{p^{i_{l-1}^{\prime}-t}}-\frac{\upsilon_{1}}{p^{i_{1}-t}}-\cdots-\frac{\upsilon_{a}}{p^{i_{a}-t}}>0,$ we get that

[TABLE]

Therefore,

[TABLE]

a contradiction.

• •

Suppose that $\frac{\upsilon_{1}}{p^{i_{1}^{\prime}-t}},\cdots,\frac{\upsilon_{l-1}}{p^{i_{l-1}^{\prime}-t}}$ appear as sumands in $\frac{\upsilon_{1}}{p^{i_{1}-t}}+\cdots+\frac{\upsilon_{a}}{p^{i_{a}-t}}$, and $\frac{\omega_{1}}{p^{j_{1}^{\prime}-t}},\cdots,\frac{\omega_{r-1}}{p^{j_{r-1}^{\prime}-t}}$ appear as sumands in $\frac{\omega_{1}}{p^{j_{1}-t}}+\cdots+\frac{\omega_{b}}{p^{j_{b}-t}}.$

Then

[TABLE]

and

[TABLE]

Then we get $i_{l}=i_{l}^{\prime}+n,....,i_{a}=i_{a}^{\prime}+n$ and $j_{r}=j_{r}^{\prime}+n,...,j_{b}=j_{b}^{\prime}+n.$

That is, we get a contradiction because the term with exponent

[TABLE]

belongs to $h$ and not to $\overline{g}.$

It follows that the terms from $\varphi_{t}\overline{g}^{p^{t}}$ whose exponents satisfy in their expressions (3.7) that $n\geq k_{t},$ cannot cancel with the terms from $\varphi_{t}h^{p^{t}}$ for which the exponents satisfy that $i_{1}^{\prime},...,i_{l-1}^{\prime},i_{1},\ldots,i_{a}>t$ and $j_{1}^{\prime},...,j_{r-1}^{\prime},j_{1},...,j_{b}>t.$

ii)

Now suppose that at least one of the $i_{1}^{\prime},...,i_{l-1}^{\prime},i_{1},\ldots,i_{a}$ is less than $t$ and $j_{1}^{\prime},...,j_{r-1}^{\prime},j_{1},...,j_{b}>t.$

Without loss of generality suppose that $i_{1}^{\prime},...,i_{k}^{\prime}<t,$ $i_{k+1}^{\prime},...,i_{l-1}^{\prime},i_{1},\ldots,i_{a}>t$ and $j_{1}^{\prime},...,j_{r-1}^{\prime},j_{1},...,j_{b}>t.$

Now we can apply a similar argument as in the previous case replacing the set formed by $i_{1}^{\prime},...,i_{l-1}^{\prime},$ $i_{1},\ldots,i_{a}$ by the set formed by $i_{k+1}^{\prime},...,i_{l-1}^{\prime},i_{1},\ldots,i_{a}$ to get a contradiction.

iii)

For the case where at least one of the $j_{1}^{\prime},...,j_{r-1}^{\prime},j_{1},...,j_{b}$ is less than $t$ and $i_{1}^{\prime},...,i_{l-1}^{\prime},i_{1},\ldots,i_{a}$ are greater than $t$ and the case where at least one of the $i_{1}^{\prime},...,i_{l-1}^{\prime},i_{1},\ldots,i_{a}$ is less than $t$ and at least one of the $j_{1}^{\prime},...,j_{r-1}^{\prime},j_{1},...,j_{b}$ is less than $t$ we can apply a similar argument as in the previous cases to obtain a contradiction.

Thus the terms from $\varphi_{t}\overline{g}^{p^{t}}$ whose exponents satisfy in their expressions (3.7) that $n\geq k_{t},$ cannot cancel with the terms from $\varphi_{t}h^{p^{t}}$.

For any $t\in\{0,1,...,d\},$ let $n_{t}:=t+\mathrm{max}\{k_{t^{\prime}}-t^{\prime}\mid t^{\prime}\in\{0,...,d\}\}.$ Here $k_{t^{\prime}}$ is defined as above and we can suppose that $k_{t^{\prime}}>d.$ We define

[TABLE]

Note that for any $t\in\{0,1,...,d\},$ we can write

[TABLE]

[TABLE]

Let us write

[TABLE]

We are going to see that the terms from $\psi^{\prime\prime}$ cannot cancel with the terms from $\psi^{\prime}.$

Note that the terms from $\varphi_{t}G_{t}^{p^{t}}$ are part of the terms of $\varphi_{t}\overline{g}^{p^{t}}$ whose exponents hold that $n\geq k_{t}.$ It follows that the terms from $\varphi_{t}G_{t}^{p^{t}}$ cannot cancel with the terms from $\varphi_{t}h^{p^{t}}.$ Since the exponents of the terms from $\varphi_{t}G_{t}^{p^{t}}$ have all the same form for any $t\in\{0,...,d\},$ that is, they are of the form

[TABLE]

where $m\geq\mathrm{max}\{k_{t^{\prime}}-t^{\prime}\mid t^{\prime}\in\{0,...,d\}\},$ we obtain that the terms from $\varphi_{t}G_{t}^{p^{t}}$ cannot cancel with the terms from $\varphi_{t^{\prime}}h^{p^{t^{\prime}}}$ for any $t^{\prime}\in\{0,...,d\}.$ It follows that the terms from $\psi^{\prime\prime}$ cannot cancel with the terms from $\psi^{\prime}.$

We can write

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $d_{0}:=\mathrm{max}\{k_{t^{\prime}}-t^{\prime}\mid t^{\prime}\in\{0,...,d\}\}$.

We are going to show that terms from $\psi^{\prime\prime}$ cannot cancel with terms from $-\psi^{\prime}-\psi^{\prime\prime}:$

suppose that

[TABLE]

where $m\geq d_{0}$ is equal to

[TABLE]

where $-(d-1)\leq n\leq d_{0}-1.$ If some of the $i_{l}^{\prime}+n,...,i_{a}^{\prime}+n$ is $\leq 0$ clearly we get a contradiction. Similarly we get a contradiction, if some of the $j_{r}^{\prime}+n,....,j_{b}^{\prime}+n$ is $\leq 0.$

In other case we get that $m=n\leq d_{0}-1,$ a contradiction. It follows that $\psi^{\prime\prime}=0.$ Since $G_{0},G_{1}^{p},...,G_{d}^{p^{d}}$ have support in $(-1,0]\times(-1,0],$ by Lemma 2.1 there are $c_{0},...,c_{n}\in\mathbb{F}$ not all zero such that

[TABLE]

That is,

[TABLE]

[TABLE]

[TABLE]

Note that the coefficient of $t_{1}^{-\frac{\upsilon_{l}}{p^{i_{l}^{\prime}+n}}-\cdots-\frac{\upsilon_{a}}{p^{i_{a}^{\prime}+n}}}t_{2}^{-\frac{\omega_{r}}{p^{j_{r}^{\prime}+n}}-\cdots-\frac{\omega_{b}}{p^{j_{b}^{\prime}+n}}}$ for $n\geq d_{0}$ is

[TABLE]

Thus, by Lemma 2.2 the sequence $a_{n}$ becomes eventually periodic.

We are going to consider the cases of sequences of the form $b_{n}$ and $c_{n}.$ Consider the vector $\omega=(0,1)$ and the mapping $\upsilon_{\omega}:\mathbb{F}((t_{1}^{\mathbb{Q}},t_{2}^{\mathbb{Q}}))\rightarrow\mathbb{Q}\cup\{\infty\}$ defined by

[TABLE]

For $h\in\mathbb{F}((t_{1}^{\mathbb{Q}},t_{2}^{\mathbb{Q}}))\setminus\upsilon_{\omega}^{-1}(\infty),$ we will denote by $\mathrm{In}_{\omega}(h),$ the sum of the terms of $h$ such that the exponents of these terms have second coordinate equal to $\upsilon_{\omega}(h)$.

Consider the valuation induced by $\upsilon_{\omega}$ (Remark 3.4), we will denote it by $\nu_{\omega}$. Since $f$ is algebraic over the ring of power series there is a polynomial

[TABLE]

such that $P(f)=0.$ So we can write,

[TABLE]

From this inequality it follows that the minimum happen at least twice. Then

[TABLE]

where $\Lambda$ is the set of indices $r$ such that $\upsilon_{\omega}(\varphi_{r}f^{p^{r}})$ is the minimum in (3.8 ). Note that,

$\mathrm{In}_{\omega}(f)=\sum_{\alpha_{2}=\nu_{\omega}(f)}f_{\alpha}t_{1}^{\alpha_{1}}t_{2}^{\alpha_{2}}$ and $\mathrm{In}_{\omega}(\varphi_{r})=t_{2}^{\nu_{\omega}(\varphi_{r})}A_{r}(t_{1}).$ By the equality (3.9) we get that $\sum_{\alpha_{2}=\nu_{\omega}(f)}f_{\alpha}t_{1}^{\alpha_{1}}t_{2}^{\alpha_{2}}$ is algebraic over $\mathbb{F}((t_{1},t_{2})).$ Since $f-\mathrm{In}_{\omega}(f)$ is algebraic over $\mathbb{F}((t_{1},t_{2})),$ we can repeat the above argument for $f-\mathrm{In}_{\omega}(f)$ and so on and so on to conclude that series of the form

[TABLE]

are algebraic over $\mathbb{F}((t_{1},t_{2})).$ Now we can apply Lemma 3.1 to conclude that sequences of the form $b_{n}$ becomes eventually periodic. For sequences of the form $c_{n}$ we can use a similar argument as above but now with $\omega=(1,0).$

∎

By combining Theorem 3.3 and Theorem 3.6 we get the following corollary.

Corollary 3.7**.**

Let $\mathbb{F}$ be a finite field and $s_{1},s_{2}\in\mathrm{A}_{p}.$ Consider the series

[TABLE]

Then $f$ is algebraic over $\mathbb{F}((t_{1},t_{2}))$ if and only if there exist positive integers $M$ and $N$ such that every sequence of the form $a_{n}=f_{(\frac{-s_{1}}{p^{i_{0}+n}},\frac{-s_{2}}{p^{j_{0}+n}})}$ where $i_{0},j_{0}\in\mathbb{Z}_{>0}$ has period $N$after $M$ terms and the sequences of the form $b_{n}=f_{(\frac{-s_{1}}{p^{i}},\frac{-s_{2}}{p^{n}})}$ and $c_{n}=f_{(\frac{-s_{1}}{p^{n}},\frac{-s_{2}}{p^{j}})}$ are eventually periodic.