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\vskip 1cm{\LARGE\bf
On Particular Families of Hyperquadratic \\
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Continued Fractions in Power Series Fields \\
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of Odd Characteristic
}
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\large
Alain Lasjaunias\\
Institut de Math\'ematiques de Bordeaux\\
CNRS-UMR 5251 \\
Universit\'e de Bordeaux \\
33405 Talence \\
France \\
\href{mailto:Alain.Lasjaunias@math.u-bordeaux.fr}{\tt Alain.Lasjaunias@math.u-bordeaux.fr} \\
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\begin{abstract}
We discuss the form of certain algebraic continued fractions in the
field of power series over $\F_p$, where $p$ is an odd prime number. This
leads to giving explicit continued fractions in these fields, satisfying an
explicit algebraic equation of arbitrary degree $d\geq 2$ and having an
irrationality measure equal to $d$. Our results are based on a mysterious
finite sequence of rational numbers.
\end{abstract}
\section{Introduction}\label{sec:intro}
We are concerned with continued fractions in the fields of formal power series over a finite field. For a general account on this matter the reader may consult Schmidt's article \cite{S} and also Thakur's book \cite{T}. Let $p$ be a prime number and $\F_q$ the finite field of characteristic $p$, having $q$ elements. Given a formal indeterminate $T$, we consider the ring $\F_q[T]$, the field $\F_q(T)$ and $\F_q((T^{-1}))$, here simply denoted by $\F(q)$, the field of power series in $1/T$ over the finite field $\F_q$. A non-zero element of $\F(q)$ can be written as
\[
\alpha=\sum_{i\leq i_0}u_iT^i \quad \text{ where }\quad i\in \Z,\quad u_i\in \F_q \quad \text{ and }\quad u_{i_0}\neq 0.
\]
An ultrametric absolute value is defined over this field by $\vert 0 \vert =0$ and $\vert \alpha \vert =\vert T \vert^{i_0}$ where $\vert T\vert$ is a fixed real number greater than 1. We also consider the subset $\F(q)^+=\lbrace \alpha \in \F(q) \quad \text{s.t.} \quad \vert \alpha \vert >1\rbrace$. Note that $\F(q)$ is the completion of the field $\F_q(T)$ for this absolute value.
In power series fields over a general finite field $\F_q$, where $q$ is a power of $p$, contrarily to the case of real numbers, the continued fraction expansion for many algebraic elements can be explicitly given. This phenomenon is due to the existence of the Frobenius isomorphism in these fields. A particular subset of $\F(q)$, denoted by $\H(q)$, containing certain algebraic elements called hyperquadratic, has been considered by Bluher and the author \cite{BL}. Let $t\geq 0$ be an integer and $r=p^t$, an irrational
element of $\F(q)$ is called hyperquadratic of order $t$ if it satisfies a non-trivial algebraic equation of the following form: \[uX^{r+1}+vX^r+wX+z=0 \quad \text{ where }\quad (u,v,w,z)\in (\F_q[T])^4.\] Note that a hyperquadratic element of order $0$ is simply quadratic. Quadratic power series, as quadratic real numbers, have an ultimately periodic continued fraction expansion. However $\H(q)$ contains power
series of arbitrary large algebraic degree over $\F_q(T)$ and the continued fraction expansion for various elements in this class has also been given explicitly. Note that if a hyperquadratic element has order $t$, it also has order $kt$ for all integers $k\geq 1$ \cite[p.\ 258]{BL}. The consideration of this subset of algebraic elements was first put forward in the study of Diophantine approximation, beginning with Mahler's article \cite{M}. For more on this topic and more references, the reader may also consult the survey of the author \cite{L1}.
Let us recall that, for an irrational element $\alpha \in \F(q)$, the irrationality measure is defined by:
\[\nu(\alpha)=\limsup_{\vert Q \vert \to \infty}(-\log \vert \alpha -P/Q \vert/\log \vert Q \vert),\]
where $P$ and $Q$ belong to $\F_q(T)$. Then $\nu(\alpha)$ is a real number greater or equal to $2$. By adapting a theorem on rational approximation for real numbers, due to Liouville in the 19th century, Mahler \cite{M} proved that, if $\alpha$ is an algebraic element of degree $d>1$ over $\F_q(T)$, then we have $\nu(\alpha)\in [2;d]$. Furthermore if $\alpha$ is any irrational number in $\F(q)$, having the infinite continued fraction expansion $\alpha=[a_1,a_2,\dots,a_n,\dots]$, then the irrationality measure is directly connected to the sequence of degrees of the partial quotients \cite[p.\ 214]{L1} and we have
\[\nu(\alpha)=2+\limsup_{n\geq 1}(\deg(a_{n+1})/\sum_{1\leq i\leq n}\deg(a_i)).\]
In this note, we present certain particular algebraic continued fractions in $\F(p)$ with odd $p$, which can be fully described. In Section~\ref{sec:cf}, we recall several generalities on continued fractions. In Section~\ref{sec:pkcf}, we describe a large family of algebraic continued fractions and, in Section~\ref{sec:perf-pkcf}, we exhibit two particular sub-families where the sequence of partial quotients is regularly distributed. The importance of these last continued fractions is highlighted in the last section, bringing families of hyperquadratic elements, in $\F(p)$ with odd $p$, having a prescribed algebraic degree, an explicit continued fraction expansion and an irrationality measure equal to the algebraic degree.
The reader will observe that this paper contains mainly conjectures based on computer calculations. Some results (see Theorem~\ref{thm:exist} below) were established in previous works, but the aim of this exposition is to point out several mathematical statements remaining largely mysterious and longing for clearness.
\section{Continued fractions}\label{sec:cf}
Concerning continued fractions in the area of function fields, we use classical notation, as it can be found for instance in the survey of the author \cite[pp.\ 3--8]{L6}. Throughout the paper we are dealing with finite sequences (or words), consequently we recall the following notation on sequences in $\F_q[T]$. Let $W=w_1,w_2,\ldots,w_n$ be such a finite sequence, then we set $\vert W\vert =n$ for the length of the word $W$. If we have two words $W_1$ and $W_2$, then $W_1,W_2$ denotes the word obtained by concatenation.
As usual, we let $[W]=[w_1,\dots,w_n]\in \F_q(T)$ denote the finite continued fraction $w_1+1/(w_2+1/(\dots ))$. In this formula the $w_i$, called the partial quotients, are non-constant polynomials. Still, we will also use the same notation if the $w_i$ are constant and the resulting quantity is in $\F_q$. However in this last case, by writing $[w_1,w_2,\dots,w_n]$ we assume that this quantity is well defined in $\F_q$, i.e., $w_n\neq 0,[w_{n-1},w_n]\neq 0,\ldots,[w_2,\dots,w_n]\neq 0$.
For $n\geq 0$, a continuant $X_n$ is a polynomial, in the $n$ variables $x_1,x_2,\ldots,x_n$, defined recursively by $X_0=1$, $X_1=x_1$ and $X_k=x_kX_{k-1}+X_{k-2}$ for $2\leq k\leq n$. We use the notation $\langle W\rangle$ for the continuant built from $W=w_1,w_2,\ldots,w_n$. In the sequel the $w_i's$ are in $\F_q[T]$, then the degree in $T$ of $\langle W\rangle$ is clearly equal to the sum of the degrees in $T$ of the $w_i's$.
We let $W'$ (resp., $W''$) denote the word obtained from $W$ by removing the first (resp., last) letter of $W$. Hence, we recall that we have $[W]=\langle W\rangle/\langle W'\rangle$. We let $W^*=w_n,w_{n-1},\ldots,w_1$, be the word $W$ written in reverse order. We have $\langle W^*\rangle$=$\langle W\rangle$ and also $[W^*]=\langle W\rangle/\langle W''\rangle$.
Moreover, if $y\in \F_q^*$, then we define $y\cdot W$ as the following sequence\[y\cdot W = y w_1, y^{-1}w_2,\ldots, y^{(-1)^{n-1}}w_n.\]
Then, it is also known that $\langle y\cdot W\rangle=y\langle W\rangle$ (resp., $=\langle W\rangle$) if $|W|$ is odd (resp., if $|W|$ is even) and $[y\cdot W]=y[W]$.
If $\alpha \in \F(q)$ is an infinite continued fraction, $\alpha=[a_1,a_2,\ldots,a_n,\ldots]$, we set $x_n=\langle a_1,a_2,\dots,a_n\rangle$ and $y_n=\langle a_2,\dots,a_n\rangle$. In this way, we have $x_n/y_n=[a_1,a_2,\ldots,a_n]$, with $x_1=a_1$, $y_1=1$ and by convention $x_0=1$, $y_0=0$. We introduce $\alpha_{n+1}=[a_{n+1},a_{n+2},\ldots]$ as the tail of the expansion of the complete quotient
($\alpha_1=\alpha$), and we have
\[\alpha=(x_{n}\alpha_{n+1}+x_{n-1})/(y_{n}\alpha_{n+1}+y_{n-1})\quad \text{ for }\quad n\geq 1.\]
We recall the following general result \cite[p.\ 332]{L2}.
\begin{proposition}\label{prop:general}
Let $p$ be a prime number, $q=p^s$ and $r=p^t$ with $s,t\geq 1$. Let $\ell\geq 1$ be an integer and $(a_1,a_2,\ldots,a_\ell)\in (\F_q[T])^l$, with $\deg(a_i)>0$ for $1\leq i\leq \ell$. Let $(P,Q)\in (\F_q[T])^2$ with $\deg(Q)<\deg(P)2$ and throughout this note, $k$ is an integer with $1\leq k