Oct 21, 2004 - the subset of irrational elements $\Theta$ in $\mathrm{F}(q)$ ..... We consider a sequence $(\phi)_{1\geq 1}$ .... and $q=p^{2\iota+1}$ be given ...
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DIOPHANTINE APPROXIMATION IN POSITIVE CHARACTERISTIC(Analytic Number Theory and Surrounding Areas) Lasjaunias, A.
数理解析研究所講究録 (2006), 1511: 156-165
2006-08
http://hdl.handle.net/2433/58599
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Kyoto University
数理解析研究所講究録 1511 巻 2006 年 156-165
156 DIOPHANTINE APPROXIMATION IN POSITIVE CHARACTERISTIC by A. Lasjaunias
Laboratoire Universit\’e Bordeaux I (FRANCE) Notes pour un expo\’e fait le 21-10-2004 \‘a l’Institut de Recherche en Sciences Math\’ematiques de Kyoto (Japon). $\mathrm{A}2\mathrm{X}$
Introduction and notations
1
The field
1.1
$\mathrm{F}(q)$
be a prime number and $q=p^{s}$ with a positive integer . We consider the finite field with elements. Then we introduce with an indeterminate , the ring of polynomials and the field of rational functions . We also consider the absolute value defined on by $|P/Q|=|T|^{\deg P-\deg Q}$ for $P,$ , where is a fixed real number greater than one. By completing with this absolute value we obtain a field denoted by which is the field power . in of formal series with coefficients Thus if is a non-zero element of we have
Let
$s$
$p$
$q$
$\mathrm{F}_{q}$
$T$
$\mathrm{F}_{q}[T]$
$\mathrm{F}_{q}(T)$
$\mathrm{F}_{q}(T)$
$Q\in \mathrm{F}_{q}[T]$
$|T|$
$\mathrm{F}_{q}(T)$
$\mathrm{F}(q)$
$\alpha$
$\mathrm{F}_{q}$
$\mathrm{F}(q)$
$\alpha=\sum_{k\leq h_{0}}u_{k}T^{k}$
with
$u_{k}\in \mathrm{F}_{q},u_{k_{0}}\neq 0$
and
$|\alpha|=|T|^{k_{0}}$
.
Observe the analogy between the classical construction of the field of real numbers and the field of power series which we are considering here. The r\^oles of , and are played by IFY, and respectively. Clearly the same construction as above can be made from an arbitrary base field $K$ instead , then the resulting field is called the field of power series over $K$ and of will be denoted by . We study here rational approximation to elements of which are algebraic over $K(T)$ . We are concerned with the case of $K$ having positive characteristic and mainly . For a presentation in a larger context and for more references see [13]. Indeed the finiteness of the base field plays an essential r\^ole in many results and this makes the field particularly interesting. $\{\pm 1\}$
$\mathbb{Z},$
$\mathbb{Q}$
$\mathrm{R}$
$\mathrm{F}_{q}[T],$
$\mathrm{F}_{q}(T)$
$\mathrm{F}(q)$
$\cdot$
$\mathrm{F}_{q}$
$\mathrm{F}(K)$
$\mathrm{F}(K)$
$K=\mathrm{F}_{q}$
$\mathrm{F}(q)$
157
1.2
Continued fractions in
$\mathrm{F}(q)$
As in the classical context of the real numbers, we have a continued fraction algorithm in . For a general study on this subject and more references see [10]. If we can write $\mathrm{F}(q)$
$\alpha\in \mathrm{F}(q)$
$\alpha=a_{0}+1/(a_{1}+1/(a_{2}+\ldots=[a_{0}, a_{1}, a_{2}, \ldots.]$
where
$a_{i}\in \mathrm{F}_{q}[T]$
.
The are called the partial quotients and we have $\deg a_{i}>0$ for $i>0$ . This continued fraction expansion is finite if and only if . As in the classical theory we define recursively the two sequences of polynomials and by $x_{n}=a_{n}x_{n-1}+x_{n-2}$ and $y_{n}=a_{n}y_{n-1}+y_{n-2}$ , $a_{i}$
$\alpha\in \mathrm{F}_{q}(T)$
$(y_{n})_{n\geq 0}$
$(x_{n})_{n\geq 0}$
with the initial conditions $x_{0}=a_{0},$ $x_{1}=a_{0}a_{1}+1,$ $y_{0}=1$ and $y_{1}=a_{1}$ . We have $x_{n+1}y_{n}-y_{n+1}x_{n}=(-1)^{n}$ , whence and are coprime polynomials. The . rational is called a convergent to and we have Because of the ultrametric absolute value we have $x_{n}$
$x_{n}/y_{n}$
$y_{n}$
$x_{n}/y_{n}=[a_{0}, a_{1}, a_{2}, \ldots, a_{n}]$
$\alpha$
$|\alpha-x_{n}/y_{n}|=|x_{n+1}/y_{n+1}-x_{n}/y_{n}|=|y_{n}y_{n+1}|^{-1}=|a_{n+1}|^{-1}|y_{n}|^{-2}$
1.3
A subset of algebraic elements in
$\mathrm{F}(q)$
Here we are concerned by elements in which are algebraic over we introduce a special subset of algebraic elements. Let $r=p^{t}$ where an integer. We denote by the subset of irrational elements in that there exist $B,$ $C,$ with
$\mathrm{F}_{q}(T)$
$\mathrm{F}(q)$
$H_{t}(q)$
$A,$
.
$\Theta$
and is such
$t\geq 0$
$\mathrm{F}(q)$
$D\in \mathrm{F}_{q}[T]$
$\alpha=\frac{A\alpha^{f}+B}{C\alpha^{f}+D}$
.
(1)
. If is the smallest non-negative integer such that satisfies an equation of type (1) we will say that is an hyperquadratic element of order . With our definition an hyperquadratic element of order zero is simply a quadratic element. We observe that elements of which are quadratic or cubic over . belong to are linked over since then 1, We put
$H(q)= \bigcup_{t\geq 0}H_{t}(q)$
$t$
$\alpha\in \mathrm{F}(q)$
$\alpha$
$t$
$\mathrm{F}(q)$
$H_{1}(q)$
$\mathrm{F}_{q}(T)$
$\alpha,$
$\alpha^{p},$
$\alpha^{p+1}$
$\mathrm{F}_{q}(T)$
, then by iteration in equation (1), Note that if for all positive integer . Moreover $H(q)$ contains also elements of arbitrarily large degree over . Note that if $K$ is a field of positive characteristic but not necessarily finite we will use the notation $H(K)$ for the corresponding subset of algebraic elements in . We recall that the analogue of Lagrange’s theorem on quadratic real numbers holds in the context of power series over a finite field. THEOREM 1. Let be irrational. Then the sequence of partial quotients in the continued fraction expansion of a is ultimately periodic if and only if a $\alpha\in H_{t}(q)$
$\alpha\in H_{kt}(q)$
$k$
$\mathrm{F}_{q}(T)$
$p$
$\mathrm{F}(K)$
$\alpha\in \mathrm{F}(q)$
$\in$
$H_{0}(q)$
.
158 As we will see the elements of the subset $H(q)$ have special properties of rational approximation due to the form of equation (1). For these elements the sequence of the degrees of the partial quotients can be bounded (as for instance in the case of quadratic elements) or unbounded. To illustrate this, given a prime let us introduce the element of defined by the following infinite expansion $p$
$\mathrm{F}(p)$
$\alpha_{1}=[T, T^{f}, T^{r^{2}}, \ldots, T^{t^{n}}, \ldots]$
where
$r=p^{t}$
whith $t\geq 0$ .
Because of the property of the Frobenius isomorphism, this element is indeed ) satisfying the equation and it belongs algebraic (quadratic if $H(p)$ . to $\alpha_{1}=T+1/\alpha_{1}^{f}$
$\mathrm{r}=1$
General results
2
Mahler’s Theorem
2.1
Diophantine appoximation in the function field case was initiated by K. Mahler [1]. The starting point in the study of rational approximation to algebraic real numbers is a famous theorem of Liouville established in 1850. This theorem has been adapted by Mahler in the fields of power series with an arbitrary base field.
be an algebraic THEOREM 2 (K. Mahler,1949). Let be a field and element over $K(T)$ of degree $n>1$ . Then there is a positive real number $K$
$a\in \mathrm{F}(K)$
$Csu\mathrm{c}h$
that
$|\alpha-P/Q|\geq C|Q|^{-n}$
for all
$P,$
$Q\in K[T]$
, with
$Q\neq 0$
.
In the case of real numbers, it is well known that Liouville’s theorem has been improved until Roth’s theorem which was established in 1955. This improvement can be transposed in fields of power series if the base field has characteristic zero as it was proved by Uchiyama in 1960. In this case the exponent in the right hand side of the inequality in the above theorem can be replaced by for all . But this is not the case in positive characteristic and consequently the study of rational approximation to algebraic elements becomes more complex. $n$
$2+\epsilon$
$\epsilon>0$
The approximation exponent
2.2 Let $\alpha$
$\alpha\in \mathrm{F}(K)$
be an irrational element. We define the approximation exponent of
by $\nu(\alpha)=\lim_{1Q\}arrow}\sup_{\infty}(-\frac{\log|\alpha-P/Q|}{\log|Q|})$
159 where and run over polynomials in $K[T]$ with $Q\neq 0$ . Considering the continued fraction expansion , since the convergents are the best rational. approximations to , it is clear, , that the approximation exponent can also be defined directly by $P$
$Q$
$\alpha=[a_{0}, a_{1}, \ldots , a_{n}, \ldots]$
$\mathrm{h}\mathrm{o}\mathrm{m}|\alpha-x_{n}/y_{n}|=|a_{n+1}|^{-1}|y_{n}|^{-2}$
$\alpha$
$\nu(\alpha)=2+\lim_{k}\sup(\deg a_{k+1}/\deg y_{k})$
.
Observe that $\deg y_{k}=\sum_{1\leq i1$ , we have $\nu(\alpha)$
$(\deg a_{1})_{i\geq 1}$
$a_{i})_{i\geq 1}$
$(\overline{\mathrm{d}}$
$\alpha\in \mathrm{F}(q)$
$\nu(\alpha)\in[2, n]$
$\mathrm{F}_{q}(T)$
.
In the same paper [1], K. Mahler introduced for a prime number of defined by
$p$
the element
$\mathrm{F}(p)$
with
$\alpha_{2}=\sum_{k\geq 0}T^{-\tau^{h}}$
$r=p^{t}$
and $t>0$ .
He observes that this element atisfies . Thus and has it is an algebraic element of degree . With our notations belongs to $H(p)$ . Note that, according to its continued fraction expansion, the element introduced above has. $\nu(a_{1})=r+1$ and is algebraic of degree $r+1$ . $\nu(\alpha_{2})=r$
$\alpha_{2}=1/T+\alpha_{2}^{f}$
$s$
$r$
$\alpha_{2}$
$\alpha_{1}\in H(p)$
2.3
Osgood’s Theorem
At the begining of the years $1970’ s$ , C. Osgood [2] used Differential Algebra to study diophantine approximation in the function field case. We introduce on the ordinary formal differentiation where $(aT^{n})’=anT^{n-1}$ if $\in K$ and $n\in$ Z. Observe that if $K$ has positive characteristic then the subfied of constants for this differentiation is the field of power series in over $K$ . If is algebraic of degree over $K(T)$ then we have $P(\alpha)=0$ where $P\in K[T][X]$ and differentiating this equation we obtain . Therefore and consequently there is an integer with $0\leq d\leq n-1$ such that where $Q\in K(T)[X]$ and $\deg_{X}(Q)=d$ . If $d\leq 2$ then we say that satisfies a Riccati differential equation. Then C. Osgood was able to prove: $\mathrm{F}(K)$
$a$
$p$
$T^{\mathrm{p}}$
$\alpha\in \mathrm{F}(K)$
$n$
$\alpha’P_{X}’(\alpha)+P_{T}’(\alpha)=0$
$\alpha’\in K(\alpha, T)$
$d$
$\alpha’=Q(\alpha)$
$\alpha$
THEOREM 3 (C. Osgood,1974). Let $K$ be a field of positive characteristic. Let be an algebraic element over $K(T)$ , of degree $n>1$ . Then if does not satisfy a Riccati differential equation there is a positive real number such that $a\in \mathrm{F}(K)$
$a$
$C$
$|\alpha-P/Q|\geq C|Q|^{-([n/2]+1)}$
160
for all
with $Q\neq 0$ . In the same paper C. Osgood also introduced a new family of algebraic elements in with a maximal approximation exponent. If $n>1$ is an integer coprime with then defined by $\alpha_{3}^{n}=1+1/T$ is algebraic of degree over . Observe that if is the order of in and has and $r=p^{t}$ then we have and consequently . $P,$
$Q\in K[T]_{7}$
$\mathrm{F}(p)$
$p$
$\alpha_{3}\in \mathrm{F}(p)$
$n$
$\nu(\alpha_{3})=n$
$t$
$p$
$a_{3}=(1+1/T)^{-(\mathrm{r}-1)/n}\alpha_{3}^{f}$
2.4
Continued fractions in
$\mathrm{F}_{p}(T)$
$(\mathbb{Z}/n\mathbb{Z})^{*}$
$\alpha_{3}\in H(p)$
$\mathrm{H}(\mathrm{q})$
Very shortly afterwards, L. Baum and M. Sweet [3] have studied diophantine approximation in by mean of the continued fraction expansion. They gave different examples of algebraic continued fractions with bounded or unbounded partial quotients. Particularly they could prove the following: THEOREM 4 (L. Baum and M.Sweet,1976). Let be the unique root in of algebraic the irreducible equation $\mathrm{F}(2)$
$\mathrm{F}(2)$
$\alpha$
$Ta^{3}+\alpha+T=0$
then the partial quotients of have degree one or two. Several years later (1986), this important work by L. Baum and M. Sweet on algebraic continued fractions in was deeply extended by W. Mills and D. Robbins [4]. They first pointed at the r\^ole played by the shape of the equation. In arbitrary positive characteristic they introduced the elements of the subset we have denoted above by $H(q)$ . Moreover they developped an algorithm to obtain the continued fraction expansion for such elements. Thu they could describe precisely the expansion for the cubic example introduced by L. Baum and M. Sweet. Also they could produce algebraic and nonquadratic examples in for $p\geq 3$ all with all partial quotients of degree one. Finally they proved: If $a\in H(q)$ and if in equation (1) we have $\deg(AD-BC)1$ . Assume . Then for every that there is a positive real number such that $\alpha\in \mathrm{F}(K)$
$C$
$\epsilon>0$
$\alpha\not\in H(K)$
$|\alpha-P/Q|\geq C|Q|^{-([n/2]+1+\epsilon)}$
for all
$P,$
$Q\in K[T]_{f}$
with
$Q\neq 0$
.
Using Osgood’s theorem and the hypothesis of a finite base field, we could prove almost the same result [8]: be an algebraic element THEOREM 8 (B. de Mathan,A.L.,1998). Let . Then there is a positive real over of degree $n>1$ . Assume that number such that $a\in \mathrm{F}(q)$
$\alpha\not\in H(q)$
$\mathrm{F}_{q}(T)$
$C$
$|a-P/Q|\geq C|Q|^{-([n/2]+1)}$
for all
$P,$
3
Two subclasses in
$Q\in K[T]$
, with
$Q\neq 0$
.
$\mathrm{H}(\mathrm{q})$
In spite of the attempt made by Mills and Robbins [5], the possibility of describing the continued fraction expansion for all the elements in $H(q)$ is yet out of reach. Nevertheless an explicit description is possible for many examples and al so for
large subclasses.
162 3.1
Elements of class IA
We will say that an element in $H(q)$ is of class IA if we have $AD-BC\in \mathbb{P}_{q}$ in introduced above belongs to this subclass. Observe equation (1). The number that, according to the property first stated by Mills and Robbins, if is of class IA and $r>1$ then $\deg(AD-BC)=01}$
$\mathrm{t}\overline{\mathrm{h}}\mathrm{e}$
$l$
$\epsilon$
$\lambda_{1},$
$\lambda_{l}$
$\lambda_{2},$
$\ldots,$
Solutions in characteristic two
3.2.1
THEOREM 11 (A.L.,J-J. Ruch, 2004). We assume that $q=2^{s}$ and $r=2^{t}$ with such that be given in integers and . Let two $po\mathit{8}itive$
$t$
$s$
$\lambda_{1},$
$\lambda_{l}$
$\lambda_{2},$
$\mathrm{F}_{q}$
$\ldots,$
$(H_{f})$
$x_{r,\mathrm{t}-r}T^{-f+1}\in \mathbb{P}_{q}$
.
satisfying the system in Then there is a sequence $n\geq 0$ by the following formulas is defined for $(\lambda_{n})_{n\geq 1}$
$\mathbb{P}_{q}$
$(S)$
. This sequence
$(\lambda_{n})_{\mathrm{n}\geq 1}$
$\lambda_{\mathrm{r}n+l+1}=\epsilon^{(-1)^{n}}\lambda_{n+1}^{\tau}\prod_{1-\mathrm{r}+2\leq i\leq \mathrm{t}}\lambda_{i}^{-1}$
$\lambda_{fn+1+k}=$ We can notice that condition Moreover is empty and 3.2.2
only involves the elements . reduces to
$(H_{f})$
$(H_{4})$
$(H_{2})$
$2\leq k\leq r$
.
$\lambda_{1-t+2},$
$\lambda_{l}$
$\ldots,$
.
$\lambda_{l}=\lambda_{1-2}$
Solutions in odd characteristic
be two sequences and We introduce the following definition. Let in . Given two integers and with $l\geq r\geq 3$ , we say that the sequence if are given and for $n\geq 0$ is $(r, l)$ -derived from the sequence we have $(y_{n})_{n\geq-1}$
$(\lambda_{n})_{n\geq 1}$
$\mathbb{P}_{q}$
$r$
$(\lambda_{n})_{n\geq 1}$
$l$
$(y_{n})_{n\geq-1}$
$\lambda_{\mathrm{r}n+l+1}=(1/2)(y_{n}+\mathrm{y}_{n-1}^{-1})$
and
$\lambda_{1},$
$\lambda_{l}$
$\ldots,$
$\lambda_{fn+1+k}=y_{n}^{(-1)^{k+1}}$
for
$2\leq k\leq r$
.
164 Then we have the following theorem : THEOREM 12 (idem). We assume that is an odd prime number. Let $q=p^{s}$ , $r=p^{t}$ and $l\geq r$ be given with $s\geq 1$ and $t\geq 1$ . Let and be given in such that $p$
$\lambda_{1},$
$\ldots,$
$\lambda_{l-r+1}$
$\epsilon$
$\mathbb{P}_{q}$
$(H)$
We put Let
$\epsilon=[2\lambda_{1}, -2\lambda_{2}, \ldots, (-1)^{l-f}2\lambda_{l-\mathrm{r}+1}, (-1)^{l-r+1}]^{-f}$
.
for $l-r+2\leq i\leq l$ . be the sequence in defined by the initial conditions
$\lambda_{i}=1$
$(y_{n})_{n\geq-1}$
$\mathbb{P}_{q}$
$y_{-1}=1$
,
$y_{i}=2\epsilon^{(-1)}\lambda_{i+1}^{\tau}-y_{1-1}^{-1}:\cdot$
,
$0\leq i\leq l-r$
and for $n\geq-1$ by the recursive formula $y_{n\tau+i+k}=y_{n}^{(-1)^{k_{f}}}\epsilon^{(-1)^{nr+l+k}}$
,
$0\leq k\leq r-1$
.
in which is $(r, l)$ -derived from the sequence Then the sequence satisfies the system $(S)$ . and tuples in It is possible to check that for all $l\geq r$ there exist satisfying condition $(H)$ in the above theorem. $(\lambda_{n})_{n\geq 1}$
$\mathbb{P}_{q}$
$(y_{n})_{n\geq-1}$
$(\mathbb{P}_{q})^{1-\Gamma+1}$
$\epsilon\in \mathrm{F}_{q}$
,
$(\lambda_{1}, \ldots, \lambda_{l-r+1})$
3.2.3
Singular solutions in certain flelds of odd characteristic
THEOREM 13 (idem). We assume that
$p$
is an odd prime number. Let
$r=p^{\mathrm{t}}$
and $q=p^{2\iota+1}$ be given with $s\geq 1$ and $t\geq 1$ . and $e^{f}+e+1\neq 0$ . be such that Let . and We put for $2\leq i\leq r$ . We put $u_{-1}=e^{-f}$ and be the sequence in Let defined by the initial condition $n\geq-1$ and $0\leq k\leq r-1$ by for $e\not\in \mathrm{F}_{p}$
$e\in \mathbb{P}_{q}$
$\lambda_{1}=(\mathrm{e}^{f}+e+1)^{r^{2}}(2e)^{-1}$
$(u_{n})_{n\geq-1}$
$\epsilon=e^{r-1}$
$\lambda_{i}=1$
$\mathbb{P}_{q}$
$u_{f(n+1)+k}=( \frac{u_{n}}{(1+ku_{n})(1+(k+1)u_{n})})’$
Let
for
be the sequence in and $0\leq k\leq r-1$ by
$(y_{n})_{n\geq-1}$
$n\geq-1$
$\mathbb{P}_{q}$
defined
by the initial condition
$y_{\mathrm{r}(n+1)+k}=y_{n}^{(-1)^{k_{f}}}\epsilon^{(-1)^{n+h+1}}(1+u_{n}(1+ku_{n})^{-1})$
$y_{-1}=1$
and
.
which is ( r)-denived from the sequence Then the sequ ence satisfies the system $(S)$ with $l=r$ . identically zero and $l=r$ Observe that when considering the sequence the solution in Theorem 13 becomes the one given in Theorem 12. Moreover the cardinality of the base field is important to ensure the existence of a sequence defined as above. In particular and for instance these solutions do not exist in a prime field. $(\lambda_{n})_{n\geq 1}in\mathbb{P}_{q}$
$(y_{n})_{n\geq-1}$
$r,$
$(u_{n})_{n\geq-1}$
$q=p^{2\epsilon+1}$
$(u_{n})_{n\geq-1}$
165
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fields of positive
characteristic,
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[4] W. Mills and D. Robbins, Continued fractions series, J. Number Theory 23, 388-404 (1986).
for certain
algebraic power
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fractions
[10] W. Schmidt, On continued fractions and diophantine approximation in power series fields, Acta Arith. 95, 139-165 (2000).
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fiel& of power
