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On the approximation exponent of some hyperquadratic power series.

1 introduction

Let p be a given prime number and K be a finite field of characteristic p. We denote by K[X] the ring of polynomials with coefficients in K and K(X) the field of fractions of K[X]. Let K(([X.sup.-1])) be the field of formal power series:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [alpha] = [summation] [u.sub.i][X.sup.i] be any formal power series, we define its polynomial part, denoted [[alpha]], by [[alpha]] := [summation over (i[greater than or equal to]0)] [u.sub.i][X.sup.i]. If [alpha] [not equal to] 0, then the degree of [alpha] is deg([alpha]) =

sup{i : [u.sub.i] [not equal to] 0} and deg(0) = -[infinity]. Thus, we define the not archimedean absolute value over K(([X.sup.-1])) by [absolute value of [alpha]] = [[absolute value of X].sup.deg([alpha])] where [absolute value of X] > 1, and [absolute value of 0] = 0.

There is a strong analogy between the classical construction of the field of real numbers and the field of power series which we are considering here. The role of {[+ or -] 1}, Z, Q and R are played by [K.sup.*], K[X], K(X) and K(([X.sup.-1])). As in the classical context of real numbers, we have a continued fraction algorithm in K(([X.sup.-1])). If [alpha] [member of] K(([X.sup.-1])) we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [a.sub.i] [member of] K[X]. The [a.sub.i] are called the partial quotients and we have deg [a.sub.i] > 0 for i > 0. This continued fraction is finite if and only if [alpha] [member of] K(X). We define two sequences of polynomial ([P.sub.n]) and ([Q.sub.n]) by [P.sub.0] = [a.sub.0], [Q.sub.0] = 1, [P.sub.1] = [a.sub.0][a.sub.1] + 1,

[P.sub.n] = [a.sub.n][P.sub.n-1] + [P.sub.n-2], [Q.sub.n] = [a.sub.n][Q.sub.n-1] + [Q.sub.n-2].

[P.sub.n]/[Q.sub.n] = [[a.sub.0], [a.sub.1], [a.sub.2], ..., [a.sub.n]] is called the nth--convergent of a and we have [P.sub.n][Q.sub.n-1] - [P.sub.n-1][Q.sub.n] = [(-1).sup.n-1]. Further, we have the following important equality

[absolute value of [alpha] - [P.sub.n]/[Q.sub.n]] = [[absolute value of [a.sub.n+1]].sup.-1][[absolute value of [Q.sub.n]].sup.-2]. (*)

One of the basic question in Diophantine approximation is how the irrational elements of K(([X.sup.-1])) can be approximated by rational elements. Our aim is to study the irrationality exponent of power series that are algebraic over the field of rational functions. In order to measure the quality of rational approximation, we introduce the following notation and definition. Let a be an irrational element of K(([X.sup.-1])).

For all real numbers y, we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where P and Q run over polynomials in K[X] with Q [not equal to] 0. Now the approximation exponent of a is defined by

v([alpha]) = sup{[mu] [member of] R : B([alpha], [mu]) < [infinity]}.

Note that if [P.sub.n]/[Q.sub.n] is a convergent to a then the equality (*) gives that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since the best rational approximation to [alpha] are its convergents, with the above notation, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is clear that the approximation exponent can be determined when the continued fraction of the element is explicitly known. Since [absolute value of [Q.sub.n][alpha] - [P.sub.n]] [less than or equal to] [[absolute value of [Q.sub.n]].sup.-1], for all irrational [alpha] [member of] K(([X.sup.-1])) we have v([alpha]) [greater than or equal to] 1. Furthermore Mahler's version of Liouville's Theorem says that if [alpha] [member of] K(([X.sup.-1])) is algebraic over K(X) of degree n > 1 then B([alpha], n - 1) = 0. Consequently, for [alpha] [member of] K(([X.sup.-1])) algebraic over K(X) of degree n > 1 we have v([alpha]) [member of] [1, n - 1].

We say that [alpha] [member of] K(([X.sup.-1])) is badly approximable by rational elements, which is equivalent to saying that a admits bounded partial quotients if v([alpha]) = 1 and B([alpha], 1) [not equal to] 0. We also say that [alpha] [member of] K(([X.sup.-1])) is well approximable by rationales, which is equivalent to saying that [alpha] admits unbounded partial quotients if v([alpha]) > 1. The reader who is interested in a survey on the different contributions to this topic and for full references can consult for example [2], [9] and [[10], Chap. 9].

We define now a specific class of power series noted by H, which is called the class of hyperquadratic. It contains the irrational elements [alpha] in K(([X.sup.-1])), satisfying an algebraic equation of the form

x = [[Ax.sup.r] + B]/[[Cx.sup.r] + D] (1.1)

where A, B, C, D [member of] K[X] and r = [p.sup.t], t [greater than or equal to] 0. A famous example of well approximable element in [F.sub.p](([X.sup.-1])) is given by K. Mahler [3] in 1949, which belongs to H, and satisfies the algebraic equation [alpha] = [X.sup.-1] + [[alpha].sup.p].

It gradually became apparent that the elements of class H deserve special consideration. Rational approximation of elements of class H has been studied also by J. Voloch [11] and more deeply by B. de Mathan [4]. They could show that if the partial quotients in the continued fraction expansion of such elements a are unbounded, then v([alpha]) > 1. By the work of B.de Mathan [4], we know moreover that for elements of class H, the approximation exponent v([alpha]) is a rational number and B(a, v([alpha])) [not equal to] 0, [infinity]. The possibility of describing the two subsets of H, formed on the one hand by badly approximable elements and on the other hand by well approximable elements remain open.

Now we will show how it is possible in some cases to compute the approximation exponent for an algebraic element, without knowing the whole continued fraction. This will be possible if this approximation exponent is large enough, that is to say not close to 1. A. Lasjaunias [2] has given applications to algebraic elements which are of class H and also to others which are not. The basic idea in the following result is due to J. Voloch [11]. We state below an improved version derived from B. de Mathan [5].

Theorem 1.1. Let [alpha] [member of] K(([X.sup.-1])). Assume that there is a sequence [([P.sub.n], [Q.sub.n]).sub.n[greater than or equal to]0], with [P.sub.n], [Q.sub.n] [member of] K[X], satisfying the following conditions:

(1) There are two real constants [lambda] > 0 et [mu] > 1, such that

[absolute value of [Q.sub.n]] = [lambda][[absolute value of [Q.sub.n-1]].sup.[mu]] and [absolute value of [Q.sub.n]] > [absolute value of [Q.sub.n-1]] for all n [greater than or equal to] 1.

(2) There are two real constants [rho] > 0 and [gamma] > 1 + [square root of [mu]], such that

[absolute value of [alpha] - [P.sub.n]/[Q.sub.n]] = [rho][[absolute value of [Q.sub.n]].sup.-[gamma]] for all n [greater than or equal to] 0.

Then we have v([alpha]) = [gamma] - 1. Further, if gcd([P.sub.n], [Q.sub.n]) = 1 for n [greater than or equal to] 0, we have B([alpha], v([alpha])) = [rho], and if the sequence [(gcd([P.sub.n], [Q.sub.n])).sub.n[greater than or equal to]0] is bounded then B([alpha], v([alpha])) [not equal to] 0, [infinity].

In this work, we consider an irreducible equation of the form

[Cx.sup.r] - [Ax.sup.r-1] - 1 = 0 (1)

where r > 2 is a power of p, A and C are nonzero polynomial with coefficients in K such that deg A > deg C. Note that the case C [member of] [K.sup.*] was studied by W. Schmidt (see [9]. p 158). This equation has a unique solution of strictly positive degree, furthermore, if we note by a this solution then [[alpha]] = [A/C] (see [8]. p 243). Note that the equation (1) satisfied by a can be written as x = [Ax.sup.r]/([Cx.sup.r] - 1), so [alpha] is an hyperquadratic power series and its approximation exponent belongs to [1, r - 1].

We are interested on computing the approximation exponent of [alpha]. Further, we describe its continued fraction expansion when C divides A. For this, we recall the following notations. Let [P.sub.n]/[Q.sub.n] [member of] K(X) such that [P.sub.n]/[Q.sub.n] := [[a.sub.1], [a.sub.2], ..., [a.sub.n]]. For all x [member of] K(X), we will note

[[[a.sub.1], [a.sub.2], ..., [a.sub.n]], x] := [P.sub.n]/[Q.sub.n] + 1/x

We state now a basic and technical Lemma concerning continued fractions. The idea involved in this Lemma appears for the first time in works of M. Mendes France [6] on finite continued fraction in the context of real numbers.

Lemma 1.2. Let [a.sub.1], ..., [a.sub.n] x [member of] [F.sub.q](X). We have the following equality:

[[[a.sub.1], [a.sub.2], ..., [a.sub.n]], x] = [[a.sub.1], [a.sub.2], ..., [a.sub.n], y], where y = [(-1).sup.n-1][Q-.sup.2.sub.n]x - [Q.sub.n-1][Q.sup.-1.sub.n].

Particularly we have

[[[a.sub.1], [a.sub.2]],x ] = [[a.sub.1], [a.sub.2], y], where y = -[a.sup.-2.sub.2]x - [a.sup.-1.sub.2].

The proof of this Lemma can be found in Lasjaunias's paper [1].

2 Results

Theorem 1.1 allows us to compute the approximation exponent of well approximate formal series. We obtain as application of this Theorem, interesting results for the approximation exponent of the solution of the equation (1) by giving a precise value of the exponent.

Theorem 2.1. Let [alpha] be the irrational solution of equation (1) such that gcd (A, C) = 1.

Suppose that [absolute value of [alpha]] = [[absolute value of C].sup.s] with s > [square root of r]/r - [square root of r] - 1. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof.

We consider the following sequence: [P.sub.0] = A, [Q.sub.0] = C and for n [greater than or equal to] 1

[P.sub.n] = [AP.sup.r.sub.n-1]

[Q.sub.n] = [CP.sup.r.sub.n-1] + [Q.sup.r.sub.n-1].

Then for all n [greater than or equal to] 0:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We show by recursion that for all n [greater than or equal to] 0:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Secondly, we have for all n [greater than or equal to] 1 [Q.sub.n] = [CP.sup.r.sub.n-1] + [Q.sup.r.sub.n-1] and since [absolute value of [P.sub.n-1]] = [[absolute value of C].sup.s][absolute value of [Q.sub.n-1]] then

[absolute value of [Q.sub.n]] = [[absolute value of C].sup.sr+1][[absolute value of [Q.sub.n-1]].sup.r].

Again by recursion we show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So we obtain for all n [greater than or equal to] 0:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.2)

We can verifies that if s > [square root of r]/r - [square root of r] - 1 then sr + 1/s + 1 > 1 + [square root of r]. Hence by Theorem 1.1 we conclude that v([alpha]) = s(r - 1)s + 1. Further, since gcd(A, C) = 1 then gcd([P.sub.n], [Q.sub.n]) = 1 for all n [greater than or equal to] 0 and so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that, in this Theorem, the condition [absolute value of [alpha]] = [[absolute value of C].sup.s] together with s > [square root of r]/r - [square root of r] - 1 are obtained if (r - [square root of r] - 1) deg A > (r - 1) deg C.

Example 2.1. Let [alpha] [member of] [F.sub.5](([X.sup.-1])) be the irrational solution of strictly positive degree of the equation:

([X.sub.2] - 2)[[alpha].sup.5] + [X.sup.5][[alpha].sup.4] - 1 = 0.

We have [absolute value of [alpha]] = [[absolute value of X].sup.3] = [[absolute value of [X.sup.2] - 2].sup.3/2]. Since 3/2 > [square root of 5]/5 - [square root of 5 - 1], then we get that v([alpha]) = 12/5.

Now, we will see the case when C divides A. For this case, we will give explicitly the continued fraction expansion for the solution of the equation (1). Knowing all the partial quotients of the solution of (1), we can compute the exact value of its approximation exponent.

Theorem 2.2. Let a be the irrational solution of the equation (1). Assume that C divides A. Then

[alpha] = [[[alpha].sub.0], ..., [a.sub.n], ...]

where [a.sub.0] = A/C and for all n [greater than or equal to] 0:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Furthermore, v([alpha]) = r - 1.

Proof. It is clear that if C divides A then the first partial quotient of [alpha] is [a.sub.0] = A/C and [alpha] = [a.sub.0] + 1/[[alpha].sub.1]. If [alpha] is a solution of (1) then we have

[[alpha].sup.r] = [alpha]/-A + C[alpha] = [a.sub.0][[alpha].sub.1] + 1/C.

Then C[[alpha].sup.r] = [[alpha].sub.0][[alpha].sub.1] + 1. So [Ca.sup.r.sub.0] + C/[[alpha].sup.r.sub.1] + 1. This implies that

[[[Ca.sup.r-1.sub.0], -[a.sub.0]], [a.sub.0][a.sup.r.sub.1]/C] = [[alpha].sub.1]

then, from Lemma 1.2, we obtain that [[Ca.sup.r-1.sub.0], -[a.sub.0], [[alpha].sub.3]] = [[alpha].sub.1]. Hence [a.sub.1] = [Ca.sup.r-1.sub.0] (since deg [Ca.sup.r-1.sub.0] > 0) and [a.sub.2] = -[a.sub.0] and [[alpha].sub.3] = -[[alpha].sub.0][[alpha].sup.r.sub.1]/[a.sup.2.sub.0]C + 1/[a.sub.0]. We apply again the same reasoning and we obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Further

[[alpha].sub.5] = [C/[a.sub.0]][a.sup.r.sub.2] + C/[a.sub.0][[alpha].sup.r.sub.3] = -[Ca.sup.r-1.sub.0], - [[alpha].sub.0], [[alpha].sub.7]],

So

[[alpha].sub.5] = [[-[Ca.sup.r-1.sub.0], -[a.sub.0]]. [a.sub.0][[alpha].sup.r.sub.3]/C] = [-[Ca.sup.r-1.sub.0], - [a.sub.0], [[alpha].sub.7]],

with [[alpha].sub.7] = -[[alpha].sub.0][a.sup.r.sub.3]/[a.sup.2.sub.0]C + 1/[a.sub.0]. Then [a.sub.5] = -[Ca.sup.r-1.sub.0], [a.sub.6] = -[a.sub.0]. Further

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then again

[[alpha].sub.9] = [[[Ca.sup.r-1.sub.0], -[a.sub.0]], [a.sub.0][[alpha].sup.r.sup.5]/C] = [[Ca.sup.r-1.sub.0], - [a.sub.0], [[alpha].sub.11],

with [a.sub.9] = [Ca.sup.r-1.sub.0], [a.sub.10] = -[a.sub.0] and [[alpha].sub.11] = -[[alpha].sup.r.sub.5]/[a.sub.0]C + 1/[a.sub.0]. This gives that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so one.

In general by an easy recurrences on k [greater than or equal to] 1, we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We come now to compute the approximation exponent of [alpha]. Set [u.sub.m] = deg [a.sub.m], [lambda] = deg [a.sub.0] and [mu] = deg C. So we have for k [greater than or equal to] 1

[u.sub.2k] = [lambda]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that we will obtain this approximation exponent by computing

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and for this, we will follow [[9]. p 158] fairly closely.

We have for odd k, [u.sub.2k+1] = [lambda]([r.sup.l] - [r.sup.l-1] - ... - r - 1) + [mu]([r.sup.l-1] - [r.sup.l-2] - ... - r - 1) when [2.sup.l-1] [parallel] (k + 1). For a given n, n + 1 = [2.sup.t] with t > 0. We have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore as n runs through the numbers [r.sup.t] - 1(t = 1, 2, ...), then

lim sup([u.sub.n]/[summation over (0[less than or equal to]i[less than or equal to]n-1)][u.sub.i]) = r - 2

and then v([alpha]) = 1 + r - 2 = r - 1.

We conclude this work by giving a sufficient condition on A and C to obtain a solution [alpha] of (1) with v([alpha]) > 1, without giving the exact value of v([alpha]). In fact, in [[7]. p 403], it has been proved that if an hyperquadratic element satisfying an equation of the type (1.1) (with AD - BC = [DELTA]), has a partial quotient other than the first with degree > deg [DELTA]/r - 1, then it will have unbounded partial quotients. According to the equation (1), we have deg [DELTA] = deg A.

Theorem 2.3. Let [alpha] be the irrational solution of (1). Suppose that there exist H and D [member of] K[X] such that DA - [H.sup.r]C = 1 and [absolute value of H] = [absolute value of A/C]. Suppose moreover that deg([A/C] - H) < (r - 2)deg A/r(r - 1). Then [alpha] admits unbounded partial quotients.

Proof. Let [A/C] = S then [absolute value of H] = [absolute value of S] = [absolute value of [alpha]]. We have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consider that [alpha] = S + [u.sub.1][X.sup.-1] + [u.sub.2][X.sup.-2] + ... and S - H = T.

[alpha]/h = S + [u.sub.1][X.sup.-1] + .../H = H + T + [u.sub.1][X.sup.-1] + .../H.

So if T [not equal to] 0 then [[absolute value of 1 - [alpha]/H].sup.r] = [[absolute value of T].sup.r]/[[absolute value of H].sup.r]. Consequently, since [absolute value of DA] = [absolute value of [H.sup.r]C] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since deg T < (r - 2)deg A/r(r - 1) then there exists a partial quotient of [alpha] of degree deg A - r deg T > deg A/r - 1. So we conclude that a admits unbounded partial quotients. Now if T = 0. Suppose that [u.sub.1] [not equal to] 0 then [[absolute value of - [alpha]/H].sup.r] = 1/[[absolute value of X].sup.r][[absolute value of H].sup.r] So we obtain:

[absolute value of [alpha] - [H.sup.r]/D] = [1/[[absolute value of D].sup.2]][1/[[absolute value of X].sup.r][absolute value of A]].

Hence, there exists a partial quotient of a of degree deg A + r > deg A/r - 1.

So we conclude that [alpha] admits unbounded partial quotients.

Example 2.2. Let Let a be the irrational solution of the equation (1) with K = [F.sub.3]/ r = 3/ A = [X.sup.4] + 2[X.sup.2] + 1 and C = [X.sup.3]. Then a admits unbounded partial quotients.

For this example, there exists H = [A/C] = X and D = [X.sup.2] + 1 such that DA - [H.sup.3]C = 1, so by the previous Theorem we conclude that [alpha] admits unbounded partial quotients.

Acknowledgement

I would like to thank Alain Lasjaunias for helpful discussions.

References

[1] A. Lasjaunias, Continued fractions for hyperquadratic power series over finite field, Finite Fields Appl, 14(2008), 329-350.

[2] A. Lasjaunias, A Survey of Diophantine Approximation in Fields of Power Series, Monatsh. Math, 130(2000), 211-229.

[3] K. Mahler, On a theorem of Liouville in fields of positive characteristic, Canadian J Math, 1(1949), 397-400.

[4] B. de Mathan, Approximation exponents for algebraic functions, Acta Arithmetica, 60(1992), 359-370.

[5] B. de Mathan, Irrationality Measures and Transcendance in Positive Characteristic, Journal of Number Theory, 54(1995), 93-112.

[6] M. Mendes France, Sur les fractions continues limitees, Acta Arith, 23(1973), 207-215.

[7] W. Mills, D. Robbins, Continued fractions for certain algebraic power series, Journal of Number Theory, 23(1986), 388-404.

[8] M. Mkaouar, Sur les fractions continues des series formelles quadratiques sur [F.sub.q](X), Acta Arithmetica, 97(2001), 241-251.

[9] W. Schmidt, On continued fractions and diophantine approximation in power series fields, Acta Arithmetica, 95 (2000), 139-166.

[10] D. Thakur, Function Field Arithmetic, World Scientific, 2004.

[11] J. F. Voloch, Diophantine approximation in positive characteristic, Periodica Mathematica Hungarica, 19(3)(1988), 217-225.

Departement de mathematiques, Faculte des sciences, University de Sfax, BP 802, 3038 Sfax, Tunisie

E.mail: ayedikhalil@yahoo.fr

Received by the editors in October 2014.

Communicated by A. Weiermann.

2010 Mathematics Subject Classification : 11J61, 11J70.
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