1.1. The field $\mathbb {K}$ of formal power series over a finite field K

Let K be a finite field with q elements. We denote the ring of polynomials with coefficients in K by $K[x]$ and the quotient field of $K[x]$ by $K(x)$ . A non-Archimedean absolute value $|\cdot |$ is defined on $K(x)$ by setting

$$ \begin{align*}|0|=0 \quad\text{and}\quad\bigg|\frac{a(x)}{b(x)}\bigg|= q^{\deg(a)-\deg(b)}, \end{align*} $$

where $a(x)$ and $b(x)$ are nonzero polynomials in $K[x]$ and $\deg (a)$ denotes the degree of $a(x)$ . The completion of $K(x)$ with respect to this absolute value is called the field of formal power series over K and is denoted by $\mathbb {K}$ . We uniquely extend the absolute value from $K(x)$ to $\mathbb {K}$ and use the same notation $|\cdot |$ . Any nonzero element $\xi $ of $\mathbb {K}$ is written uniquely as

$$ \begin{align*}\xi=\sum_{n=r}^{\infty}a_nx^{-n},\end{align*} $$

where $a_n \in K $ for $n=r, r+1,\ldots $ with $ a_r\neq 0$ and r is the rational integer satisfying $|\xi |=q^{-r}$ . The elements of $\mathbb {K}$ are called formal power series.

Let $P(y)=a_0+a_1y+\cdots +a_ny^n$ be a nonzero polynomial with coefficients in $K[x]$ . The height $H(P)$ of $P(y)$ is defined as $H(P)=\max \{|a_0|, |a_1|,\ldots ,|a_n|\}$ and the degree of $P(y)$ is denoted by $\deg (P)$ . An element $\xi $ of $\mathbb {K}$ is called an algebraic formal power series if it is algebraic over $K(x)$ and $\xi $ is called a transcendental formal power series otherwise. Let $\alpha $ be an algebraic formal power series and $P(y)$ be its minimal polynomial over $K[x]$ . Then the height $H(\alpha )$ and the degree $\deg (\alpha )$ of $\alpha $ are defined by $H(P)$ and $\deg (P)$ , respectively. The roots of $P(y)$ are called the conjugates of $\alpha $ over  $K(x)$ .

1.2. Mahler’s and Koksma’s classifications in $\mathbb {K}$

In 1932, Mahler [Reference Mahler10] gave a classification of real numbers and separated transcendental real numbers into three disjoint classes called S-, T- and U-numbers. (See Bugeaud [Reference Bugeaud2] for detailed information about Mahler’s classification of real numbers.) Bundschuh [Reference Bundschuh3] carried over Mahler’s classification to the field $\mathbb {K}$ and separated transcendental formal power series into three disjoint classes as follows.

Let $\xi $ be a transcendental formal power series. For positive rational integers n and H, define the quantities

$$ \begin{align*} w_n(H,\xi)=\min\{|P(\xi)|:P(y)\in K[x][y] \setminus \{0\},\,\deg(P)\leq n, H(P)\leq H\}, \end{align*} $$

$$ \begin{align*}w_n(\xi)=\limsup_{H\rightarrow\infty}\;\displaystyle\frac{-\log w_n(H,\xi)}{\log H}\quad\text{and}\quad w(\xi)=\limsup_{n\rightarrow\infty}\;\frac{w_n(\xi)}{n}.\end{align*} $$

Bundschuh [Reference Bundschuh3] showed that

$$ \begin{align*}w_n(H,\xi)<H^{-n}q^{n}\max\{1,|\xi|\}^n.\end{align*} $$

This implies that $w_n(\xi )\geq n$ for $n=1,2,\ldots $ and so $w(\xi )\geq 1$ . If $w_n(\xi )$ is infinite for some integer n, then denote by $\mu (\xi )$ the smallest such integer. If $w_n(\xi )$ is finite for $n =1,2,\ldots $ , put $\mu (\xi )=\infty $ . Then $\xi $ is said to be:

• • an S-number if $1\leq w(\xi )<\infty $ and $\mu (\xi )=\infty $ ;

• • a T-number if $w(\xi )=\infty $ and $\mu (\xi )=\infty $ ;

• • a U-number if $w(\xi )=\infty $ and $\mu (\xi )<\infty $ .

Further, $\xi $ is called a $U_m$ -number if $\mu (\xi )=m$ . Oryan [Reference Oryan13] gave the first explicit examples of $U_m$ -numbers in $\mathbb {K}$ . Recently, Kekeç [Reference Kekeç7, Reference Kekeç8] found further explicit examples of U-numbers in $\mathbb {K}$ by applying the method of Oryan [Reference Oryan13].

In 1939, Koksma [Reference Koksma9] gave another classification of real numbers and separated transcendental real numbers into three disjoint classes called $S^*$ -, $T^*$ - and $U^*$ -numbers. Bugeaud [Reference Bugeaud2, Section 9.4] introduced Koksma’s classification in the field $\mathbb {K}$ . According to Koksma’s classification in $\mathbb {K}$ , transcendental formal power series are separated into three disjoint classes as follows.

Let $\xi $ be a transcendental formal power series. For positive rational integers n and H, define the quantities

$$ \begin{align*}w_n^*(H,\xi)\!=\!\min\{ |\xi-\alpha|\, : \alpha \,\text{is an algebraic formal power series}, \, \deg(\alpha)\leq n, H(\alpha)\leq H\},\end{align*} $$

$$ \begin{align*}w_n^*(\xi)=\limsup_{H\rightarrow\infty}\;\displaystyle\frac{-\log (H\:w_n^*(H,\xi) )}{\log H}\quad \text{and} \quad w^*(\xi)=\limsup_{n\rightarrow\infty}\;\frac{w_n^*(\xi)}{n}.\end{align*} $$

Denote by $\mu ^*(\xi )$ the smallest integer n such that $w_n^*(\xi )=\infty $ if such an integer exists and write $\mu ^*(\xi )=\infty $ otherwise. Then $\xi $ is called:

• • an $S^*$ -number if $0< w^{*}(\xi )<\infty $ and $\mu ^{*}(\xi )=\infty $ ;

• • a $T^*$ -number if $w^{*}(\xi )=\infty $ and $\mu ^{*}(\xi )=\infty $ ;

• • a $U^*$ -number if $w^{*}(\xi )=\infty $ and $\mu ^{*}(\xi )<\infty $ .

Moreover, $\xi $ is called a $U_m^*$ -number if $\mu ^*(\xi )=m$ . Recently, Ooto [Reference Ooto12] proved that a transcendental formal power series $\xi $ in $\mathbb {K}$ is a $U_m$ -number if and only if it is a $U_m^*$ -number. (See Bugeaud [Reference Bugeaud2] for further information and references on Koksma’s classification in $\mathbb {R}$ and in $\mathbb {K}$ .)

1.3. Continued fractions in $\mathbb {K}$

Many concepts in the field $\mathbb {R}$ carry over to the field $\mathbb {K}$ . One of them is the fact that any element $\xi $ of $\mathbb {K}$ with $|\xi |\geq 1$ can be written as a continued fraction as follows.

Let $\xi $ be a formal power series with $|\xi |\geq 1$ . Then $\xi $ is written uniquely as

$$ \begin{align*}\xi=a_mx^m+a_{m-1}x^{m-1}+\cdots+a_1x+a_0+a_{-1}x^{-1}+a_{-2}x^{-2}+\cdots,\end{align*} $$

where $a_i\in K$ for $i=m,m-1,\ldots $ with $a_m\neq 0$ and m is the nonnegative rational integer satisfying $|\xi |=q^{m}$ . Put

$$ \begin{align*}\xi=[\xi]+(\xi),\end{align*} $$

where

$$ \begin{align*}[\xi]:=a_mx^m+a_{m-1}x^{m-1}+\cdots+a_1x+a_0,\quad (\xi):=a_{-1}x^{-1}+a_{-2}x^{-2}+\cdots.\end{align*} $$

Let $b_0:=[\xi ]\in K[x]$ . Note that $|b_0|=|\xi |\geq 1$ . If $(\xi )=0$ , stop. If $(\xi )\neq 0$ , write

$$ \begin{align*}\xi=b_0+\frac{1}{\xi_1},\end{align*} $$

where $\xi _1=1/(\xi ) \in \mathbb {K}$ with $|\xi _1|>1$ . Let $b_1:=[\xi _1]\in K[x]$ . Note that $|b_1|=|\xi _1|>1$ . If $(\xi _1)=0$ , stop. If $(\xi _1)\neq 0$ , write

$$ \begin{align*}\xi=b_0+\frac{1}{b_1+\dfrac{1}{\xi_2}},\end{align*} $$

where $\xi _2=1/(\xi _1)\in \mathbb {K}$ with $|\xi _2|>1$ . Let $b_2:=[\xi _2]\in K[x]$ . Note that $|b_2|=|\xi _2|>1$ . If $(\xi _2)=0$ , stop. If $(\xi _2)\neq 0$ , continue in this way. Thus, we get the unique representation

$$ \begin{align*}\xi=\langle b_0, b_1,\ldots,b_{n-1},\xi_n\rangle :=b_0+\frac{1}{b_1+\dfrac{1}{\ddots+\dfrac{1}{b_{n-1}+\dfrac{1}{\xi_n}}}},\end{align*} $$

where $b_i=[\xi _i]\in K[x]$ with $|b_i|=|\xi _i|>1$ for $i=1,2,\ldots ,n-1$ and $\xi _{i+1}=1/(\xi _{i})\in \mathbb {K}$ for $i=0,1,\ldots ,n-1$ with $\xi _{0}:=\xi $ .

If $(\xi _n)=0$ for some n, then the continued fraction expansion of $\xi $ is finite and $\xi $ is written uniquely as

$$ \begin{align*}\xi=\langle b_0, b_1, \ldots, b_{n}\rangle :=b_0+\frac{1}{b_1+\dfrac{1}{\ddots+\dfrac{1}{b_n}}}. \end{align*} $$

A formal power series $\xi $ is in $K(x)$ if and only if its continued fraction expansion is finite.

Let $(\xi _n)\neq 0$ for all n; in other words, let the continued fraction expansion of $\xi $ be infinite. Define $p_{-1}=1$ , $p_0=b_0$ , $q_{-1}=0$ , $q_0=1$ and

$$ \begin{align*}p_n=b_np_{n-1}+p_{n-2}, \quad q_n=b_nq_{n-1}+q_{n-2} \quad (n=1,2,\ldots).\end{align*} $$

By induction on n, it is easily seen that

$$ \begin{align*}\frac{p_n}{q_n}=\langle b_0, b_1, \ldots, b_{n}\rangle \end{align*} $$

and $p_n/q_n$ is called the nth convergent of the continued fraction expansion of $\xi $ . Moreover, by induction on n, we have the following properties:

1. (1) $({\beta p_n+p_{n-1}})/({\beta q_n+q_{n-1}})=\langle b_0, b_1,\ldots ,b_{n}, \beta \rangle \quad (n=0,1,2,\ldots )$ , where $\beta \in \mathbb {K}\backslash \{0\}$ ;

2. (2) $p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1} \quad (n=0,1,2,\ldots )$ ;

3. (3) $|q_n|>|q_{n-1}| \quad (n=0,1,2,\ldots )$ ;

4. (4) $|q_n|=|b_1b_2 \cdots b_{n}| \quad (n=1,2,\ldots )$ ;

5. (5) $|p_n|=|b_0b_1b_2 \cdots b_{n}|=|b_0||q_n| \quad (n=0,1,2,\ldots )$ ;

6. (6) $\xi -({p_n}/{q_n})={(-1)^n}/({q_n(\xi _{n+1}q_n+q_{n-1})}) \quad (n=1,2,\ldots )$ ;

7. (7) $|\xi -({p_n}/{q_n})|={1}/({|b_{n+1}||q_{n}|^{2}})<{1}/{|q_{n}|^{2}} \quad (n=1,2,\ldots )$ .

The properties (3) and (7) imply that $\lim _{n\to \infty } p_n/q_n=\xi $ . So, it is meaningful to write

$$ \begin{align*}\xi=\lim_{n\to\infty} \langle b_0, b_1, b_2,\ldots, b_n\rangle =: \langle b_0, b_1, b_2,\ldots\rangle \end{align*} $$

An important result on infinite continued fractions in $\mathbb {K}$ is that the continued fraction expansion of a formal power series $\alpha $ is infinite and periodic if and only if $\alpha $ is algebraic over $K(x)$ with $\deg (\alpha )=2$ . (See [Reference Chaichana, Laohakosol and Harnchoowong4, Reference Ooto12, Reference Schmidt14] for detailed information, proofs and further results on continued fractions in $\mathbb {K}$ .)

1.4. Construction of our main results

In 1982, Alnıaçık [Reference Alnıaçık1, Theorem] gave a method to construct explicit examples of real $U_m$ -numbers by using continued fraction expansions of real algebraic numbers of degree $m>1$ . In the present paper, we establish the $\mathbb {K}$ -analogue of this result.

Theorem 1.1. Let $\alpha $ be an algebraic formal power series with $|\alpha |\geq 1$ , $\deg (\alpha )=m>1$ and continued fraction expansion

(1.1) $$ \begin{align} \alpha=\langle a_0,a_1,a_2,\ldots\rangle. \end{align} $$

Let $\{r_{j}\}_{j=0}^{\infty }$ and $\{s_{j}\}_{j=0}^{\infty }$ be two infinite sequences of nonnegative rational integers satisfying

$$ \begin{align*} 0=r_{0}< s_0< r_{1}<s_{1}<r_{2}<s_{2}< r_{3}<s_{3}< \cdots \quad\text{and}\quad r_{n+1}-s_{n}\geq2. \end{align*} $$

Denote by $p_n/q_n$ the nth convergent of the continued fraction (1.1) and assume that

(1.2) $$ \begin{align} \mathrm{(a)} \ \displaystyle\lim_{n\to\infty} \frac{\log|q_{s_n}|}{\log|q_{r_n}|}=\infty, \quad \mathrm{(b)} \ \displaystyle\limsup_{n\to\infty}\frac{\log|q_{r_{n+1}}|}{\log|q_{s_n}|}<\infty. \end{align} $$

Define $b_j\in K[x] \ (j=0,1,\ldots )$ by

(1.3) $$ \begin{align} \begin{array}{@{}ll} b_j= & \left\{ \begin{array}{@{}lll}a_j, & r_n\leq j\leq s_n \ \ \ (n=0,1,\ldots),\\ v_j, &s_n<j<r_{n+1} \ (n=0,1,\ldots), \end{array} \right. \end{array} \end{align} $$

where $v_j\in K[x]$ with $1<|v_j|\leq S_1 |a_j|^{S_2}$ are such that $\sum _{j=s_n+1}^{r_{n+1}-1}|a_j-v_j|\neq 0$ and $S_1$ and $S_2$ are fixed positive rational integers. Then the formal power series $\xi \in \mathbb {K}$ with continued fraction expansion

$$ \begin{align*}\xi=\langle b_0,b_1,b_2,\ldots\rangle \end{align*} $$

is a $U_m$ -number.

We denote by $\widehat {\mathbb {K}}$ the completion of the algebraic closure of $\mathbb {K}$ with respect to $|\cdot |$ . The height $H(P)$ of a polynomial $P(y)$ in $\widehat {\mathbb {K}}[y]$ and the height $H(\alpha )$ , the degree $\deg (\alpha )$ and the conjugates of an element $\alpha $ in $\widehat {\mathbb {K}}$ which is algebraic over $K(x)$ , are defined exactly as in Section 1.1.

In 1973, İçen [Reference İçen6, page 25] gave an effective upper bound for the usual height of a complex algebraic number dependent on other complex algebraic numbers. Our second result in the present paper is the following $\widehat {\mathbb {K}}$ -analogue of İçen’s result, which we apply to prove Theorem 1.1.

Theorem 1.2. Let L be a finite extension of degree m over $K(x)$ and $\alpha _1,\alpha _2,\ldots ,\alpha _k$ be in L. Let $\eta \in \widehat {\mathbb {K}}$ be algebraic over $K(x)$ . Assume that $F(\eta ,\alpha _1,\ldots ,\alpha _k)=0$ , where $F(y,y_1,\ldots ,y_k)$ is a polynomial in $y, y_{1},\ldots ,y_{k}$ over $K[x]$ with degree at least one in y. Then

$$ \begin{align*} \deg(\eta)\leq dm\end{align*} $$

and

$$ \begin{align*}H(\eta)\leq H^mH(\alpha_1)^{l_{1}m}\cdots H(\alpha_k)^{l_{k}m},\end{align*} $$

where d is the degree of $F(y,y_1,\ldots ,y_k)$ in y, $l_j$ is the degree of $F(y,y_1,\ldots ,y_k)$ in $y_j\ (j=1,\ldots ,k)$ and H is the maximum of the absolute values of the coefficients of $F(y,y_1,\ldots ,y_k)$ .

In Section 2, we cite some lemmas we need to prove Theorem 1.1. We prove Theorem 1.2 in Section 3 and Theorem 1.1 in Section 4.