12 Conversion Algorithm of Matrix Decomposition in Ordinary Continued Fraction
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5 Generalized Pisot Numbers and Matrix Decomposition
135
where
M(p, k) =
p3 + p2 + 3
p2 + p
2
p −p+9
·
2p2 + 2p + 2
p
1
3k + 2
0
p2 + p p3 + p2 + 3
·
1
p
0
3k + 1
1
2p2 − 6p + 6
0
Ak (p) Bk (p)
3
,
=
2
p −p+9
Ck (p) Dk (p)
0 13
Ak (p) = (27 + 9p + 33p2 + 32p3 + 8p4 + 10p5 + 4p6 )k +
+9 + 5p + 16p2 + 16p3 + 4p4 + 5p5 + 2p6 ,
Bk (p) = (18 + 36p + 12p2 + 24p3 + 8p4 + 4p5 + 2p6 )k +
+6 + 21p + 5p2 + 12p3 + 4p4 + 2p5 + p6 ,
Ck (p) = (6 + 24p + 26p2 + 8p3 + 10p4 + 4p5 )k +
+4 + 13p + 14p2 + 4p3 + 5p4 + 2p5 ,
Dk (p) = (27 + 9p + 21p2 + 8p3 + 4p4 + 2p5 )k +
+18 + 4p + 11p2 + 4p3 + 2p4 + p5 .
The program on Fig. 5.2 implements an algorithm of transition from matrix decomposition α(5) to a conventional continuous fraction.
Using the symbolic computation, we obtain
M(4, k) =
M(5, k) =
31311k + 15645
7686k + 3864
16226k + 8106
,
3983k + 2002
103647k + 51809
20526k + 10294
52248k + 26111
10347k + 5188
The value of M(5, k) is used in the given program.
Lemma 5.28 The program in Fig. 5.2 realizes the conversion algorithm of a matrix
decomposition in continued fraction.
Proof Indeed, first observe that
103647k + 51809
1017k + 319
=5+
= 5,
20526k + 10294
20526k + 10294
52248k + 26111
513k + 171
=5+
=5
10347k + 5188
10347k + 5188
and
103647k + 51809
20526k + 10294
52248k + 26111
10347k + 5188
∈ M∗ .
136
N.M. Dobrovol’skii et al.
Fig. 5.2 Describes conversion algorithm of a matrix decomposition α(5) in ordinary continued
fraction
5 Generalized Pisot Numbers and Matrix Decomposition
137
By Theorem 5.12, the matrix decomposition
∞
k=0
103647k + 51809
20526k + 10294
52248k + 26111
10347k + 5188
converges.
We observe further that the outside loop for k ∈ 0..n realizes the calculation of
the product
n
103647k + 51809 52248k + 26111
20526k + 10294
10347k + 5188
k=0
and separate the product
J
j=0
qj 1
1 0
using inner loop while r = floor DB .
The auxiliary loop for kk ∈ 1..3 allows to reduce numbers in the matrix M, if
it is possible. By Lemma 5.22, the division of all elements of a matrix by common divisor does not change the value of the matrix decomposition. Therefore,
based on Theorem 5.12 and Lemma 5.27, the given program computes the partial
quotients.
5.13 Results of Symbolic Computation
The symbolic computations based on programs in Figs. 5.1 and 5.2 show that these
programs provide the same partial quotients. The calculations using the program
based on matrix decomposition are faster.
The calculations cfki(100) give the values of 592 partial quotients, and cfki(200)
give the values of 1194 partial quotients. Since the results are presented in the matrix
form containing 40 elements in each row, the last elements of the last line may be
zero (Figs. 5.3 and 5.4).
138
N.M. Dobrovol’skii et al.
Fig. 5.3 Gives the distribution of values of partial quotients taking into account the value zero that
are not partial quotients
Fig. 5.4 The program of the
calculations of the
distribution of values of
partial quotients
5.14 Conclusion
The results of this paper show that reduced algebraic irrationalities in case of totally
real algebraic fields and generalized Pisot numbers in general case play a fundamental
role for the continued fraction expansion of algebraic irrationalities. Starting with
some index, all residual fractions are the reduced algebraic numbers in the first case
and generalized Pisot numbers in the second case.
Theorem 5.12 implies that starting with the number m0 , to calculate the next
partial quotient is sufficient to calculate the two values of the minimal polynomial
fm (x). There is a recurrence formula for calculating the next partial quotient.
Apparently, it is of interest to further study the focus conjugate to the residual
m−2
.
fraction αm around the fraction − Q
Qm−1
Consider the conjugation spectrum of irrational number α, that is, the set of all
conjugate to residual fractions. The conjugation spectrum is infinite if n > 2, and it
is a finite set if n = 2.
If we will call the rational conjugate spectrum of real algebraic numbers the set of
m−2
, then the natural question arises about its structure.
all fractions of the form − Q
Qm−1
In the quadratic case, there is a finite set of limit points for the rational conjugate
spectrum that is conjugate spectrum. What is in general case?
5 Generalized Pisot Numbers and Matrix Decomposition
139
From the results of this paper, we see that the theory of linear fractional
transformations of polynomials is closely related to the theory of linear transformations of homogeneous binary form. The second theory is simpler in many respects,
and the proof of many statements is shorter.
Such relation is not casual. Apparently, the theory of linear fractional transformations of polynomials is connected with Diophantine approximations of the first kind,
and the linear transformations of homogeneous forms are connected with Diophantine approximations of the second kind.
Acknowledgments This research was supported by the Russian Foundation for Basic Research
(Grant Nos 15-01-01540, 15-41-03262, 15-41-03263).
References
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combinatorics, pp. 142–161. Nauka, Moscow (1978)
2. Berestovskii, V.N., Nikonorov, YuG: Continued fractions, the Group GL(2, Z), and Pisot
numbers. Siberian Adv. Math. 17(4), 268–290 (2007)
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4, 211–221 (1964)
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35–65 (2015)
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6090–84 (1984)
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VINITI, Moscow, No 6091–84 (1984)
7. Dobrovol’skii, N.M.: About the modern problems of the theory of hyperbolic zeta-functions
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and Applications: Proceedings of the International Conference on Algebra, Dedicated to the
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Galois on purely periodic continued fractions. Cheb. Sb. 13(3), 47–52 (2012)
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cubic irrational. Cheb. Sb. 14(1), 34–55 (2013)
12. Dobrovol’skii, N.M., Dobrovol’skii, N.N., Balaba, I.N., Rebrova, I.Yu., Polyakova, N.S.:
Linear-fractional transformation of polynomials. In: Algebra, Number Theory and Discrete
Geometry: Contemporary Issues and Applications: Proceedings of XIII International conference. A supplementary volume, Tula, pp. 134–149 (2015)
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Centre of the Academy of Sciences of USSR (1979)
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16. Podsypanin, E.V.: A generalization of the algorithm for continued fractions related to the
algorithm of Viggo Brunn. J. Sov. Math. 16, 885–893 (1981)
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Chapter 6
On the Periodicity of Continued Fractions
in Hyperelliptic Fields
Gleb V. Fedorov
Abstract Let L be a function field of a hyperelliptic curve defined over an arbitrary
field characteristic different from 2. We construct an arithmetic of continued fractions
of an arbitrary quadratic irrationality in field of formal power series with respect to
linear finite valuation. The set of infinite valuation and finite linear valuation of L
is denoted by S. As an application, we have found a relationship between the issue
of the existence of nontrivial S-units in L and periodicity of continued fractions of
some key elements of L.
6.1 Introduction
Let K be a field of characteristic different from 2, and let f (x) ∈ K [x] be a square
free polynomial of odd degree 2s + 1, s ≥ 1. Given an irreducible polynomial h(x) ∈
of the field K (x). Suppose
K [x], we use vh to denote the corresponding valuation
√
+
that vh has two extensions to the field L = K (x)( f ), namely v−
h and vh . We set
−
S = {vh , v∞ }, where v∞ is the infinite valuation of the field L.
An elementary introduction to some of the theory of hyperelliptic curves over
finite fields of arbitrary characteristic may be found, for example, in [7].
The multiplicative group O S∗ of the ring O S of S-integer elements of L is called
the group of S-units.
The article [6] given a positive answer for two questions:
• Is there a relationship between the existence of nontrivial S-units and the periodicity of the expansion of an appropriate element of in a continued fraction?
G.V. Fedorov (B)
Mechanics and Mathematics Faculty, Moscow State University,
Moscow 119991, Russia
e-mail: glebonyat@mail.ru
G.V. Fedorov
Research Institute of System Development, Russian Academy of Sciences,
Nakhimovskii Pr. 36, Korp. 1, Moscow 117218, Russia
© Springer International Publishing Switzerland 2016
V.A. Sadovnichiy and M.Z. Zgurovsky (eds.), Advances in Dynamical Systems
and Control, Studies in Systems, Decision and Control 69,
DOI 10.1007/978-3-319-40673-2_6
141
142
G.V. Fedorov
• Is it possible to calculate a fundamental S-unit by using convergents for a continued
fraction expansion of an element of L (as in the case of a finite field of constants
studied in [2] and [3])?
In this paper, we construct the arithmetic of the general continued fractions, necessary for the proof of the main results of [6].
If f (x) is a polynomial of even degree, an arithmetic of continued fractions
with respect to infinite valuation described in articles [1] and [10]. Moreover,√the
relationship between fundamental units of the hyperelliptic field L = K (x)( f )
and continued fractions is considered there. Another approach to this case uses only
the Riemann–Roch theorem for curves and manipulations of divisors related with
continued fractions (see [4]).
6.2 Continued Fractions
Let us define Oh = {ω ∈ K (x) : vh (ω) ≥ 0} is the valuation ring of the valuation
vh of the field K (x), and ρh = {ω ∈ K (x) : vh (ω) > 0} is an ideal of the valuation
vh . We fix the set of the representative of related classes Oh by ρh , so that =
{ω ∈ K [x] : deg ω < deg h}. Then, we can consider the set
((h)) =
K ((h))
=
⎧
⎨
∞
⎩
j=m
bjh j : bj ∈
⎫
⎬
, m∈Z .
⎭
The set ((h)) is called the set of a formal power series.
Let α ∈ ((h)), then α has the form
∞
bjh j.
α=
j=m
We introduce the notation
[α] = [α]h =
0
j=m
0,
b j h j , i f m ≤ 0,
i f m > 0.
We set α0 = α and a0 = [α0 ]. For j ∈ N, we define elements α j and a j by induction
as follows: If α j − a j = 0, then
αj =
1
∈
α j−1 − a j−1
((h)), a j = [α j ].
6 On the Periodicity of Continued Fractions in Hyperelliptic Fields
143
As a result, we obtain a continued fraction, for which we use the standard brief
notation [a0 , a1 , a2 , . . .]. Note that a j can be considered as an element of the field
K (x).
We set p−2 = 0, p−1 = 1, q−2 = 1, and q−1 = 0 and define elements p j , q j ∈
K (x) by induction as
p j = a j p j−1 + p j−2 , q j = a j q j−1 + q j−2 ,
j ≥ 0,
(6.1)
then p j /q j = [a0 , a1 , a2 , . . . , a j ] is the jth convergent of α. The standard way we
can show (see [8]) that for j ≥ −1, the following relations hold
q j p j−1 − p j q j−1 = (−1) j ,
(6.2)
(−1)
,
q j α j+1 + q j−1
p j α j+1 + p j−1
α=
.
q j α j+1 + q j−1
qjα − pj =
j
(6.3)
(6.4)
By the construction, for j ≥ 1 we have vh a j = vh α j < 0. From (6.1) by induction, we easily obtain relations
j
vh q j = vh a j + vh q j−1 =
vh (ai ) ,
i=1
j
vh p j = vh a j + vh p j−1 =
vh (ai ) .
i=0
From (6.3), we have
vh q j α − p j = −vh q j+1 = −vh a j+1 − vh q j > −vh q j ,
(6.5)
or equivalently,
vh α −
pj
qj
= −vh q j+1 − vh q j > −2vh q j .
Thus, lim p j /q j = α, i. e., the convergents converge to α.
j→∞
In [3], it was shown that √
an effective connection between the nontrivial S-units
in O S and the expansion of f or of elements related to in a continued fraction is
possible only if deg h = 1.
Below, we will assume that deg h = 1.
Suppose that K (x)h is the completion of the field K (x) with respect to the valuation
vh . In the case of deg h = 1, it is easily to proof that K (x)h = K ((h))
√ = ((h)).
Since by assumption vh has two extensions to the field L = K (x)( f ), then the
144
G.V. Fedorov
field L has two embeddings into K (x)h . We fix one of this embedding, so that every
element of the field L has the unique formal power series in K ((h)).
The continued fraction [a0 , a1 , a2 , . . .] of an element α ∈ K ((h)) is finite if and
only if α ∈ K (x) (see [3], Proposition 5.1). In a standard way, we can show that if the
continued fraction [a0 , a1 , . . .] for some α ∈ K ((h)) is periodic, then α is a quadratic
irrationality. In the case of an infinite field K and valuation v∞ , the converse is not
always true (see [1]). However, in the case of a field K = Fq and deg h = 1 an
assertion holds: If α ∈ K ((h)) is a quadratic irrationality, the continued fraction for
the α is periodic (see [3]).
6.3 Some Relations with Continued Fractions
Let α is a root of the polynomial
H (X ) = λ2 X 2 + 2λ1 X + λ0 ,
where λ0 , λ1 , λ2 ∈ K [x].
(6.6)
We define α is conjugate of the element α, and d = λ21 − λ2 λ0 is the shortened
discriminant of the polynomial (6.6). We assume that d/ f is a perfect square in the
field K (x), i. e., α ∈ L. Let α = [a0 , a1 , . . .] is a decomposition of α into a continued
fraction, with respect to the valuation v−
h.
For all j ≥ 0, we denote s j = −vh a j and t j = −vh q j . Since vh ( f ) = 0,
we can define t = 21 vh (d) ≥ 0, so that t ∈ Z.
By the construction of a continued fraction, for j ≥ 0 we have
j
si , vh p j = −t j − s0 .
s j ≥ 1, t j =
i=1
−
The element β ∈ L is called reduced with respect to the valuation v−
h , if vh (β) <
−
0 and vh β > 0.
Proposition 6.1 The element α + a0 is reduced if and only if vh (λ0 ) < vh (λ2 ) <
vh (λ1 ).
Proof By the construction of a continued fraction, we have v−
h (α − a0 ) > 0.
−
Assume α + a0 is reduced, then v−
h (α + a0 ) < 0 and vh (α + a0 ) > 0. By virtue
of Vieta’s formulas, we have
vh
λ1
λ2
−
= vh ((α − a0 ) + (α + a0 )) ≥ min v−
h (α − a0 ) , vh (α + a0 ) > 0,
(6.7)
from√which vh (λ2 ) < vh (λ1 ). Without loss of generality, we can assume that α =
−λ1 + d
. Since v−
h (α + a0 ) < 0, it follows that
λ2
6 On the Periodicity of Continued Fractions in Hyperelliptic Fields
vh (a0 ) =
v−
h
(α) =
v−
h
−λ1 +
λ2
√
d
145
< 0,
(6.8)
but from the inequality (6.7), we have
vh (a0 ) =
v−
h
√
d
λ2
< 0,
(6.9)
therefore
vh λ21 − λ2 λ0 = vh (d) < 2vh (λ2 ) ,
(6.10)
it means that vh (λ0 ) < vh (λ2 ).
Conversely, if vh (λ0 ) < vh (λ2 ) < vh (λ1 ), then the inequalities (6.8), (6.9), and
(6.10) hold, and by the construction of a continued fraction v−
h (α − a0 ) > 0, hence,
+
a
<
0.
Then,
we
write
v−
(α
)
0
h
−
(a0 − α) −
v−
h (α + a0 ) = vh
2λ1
λ2
≥ min v−
h (α − a0 ) , vh
2λ1
λ2
> 0,
and it was to be proved.
Let H (X, Y ) = λ2 X 2 + 2λ1 X Y + λ0 Y 2 . For j ≥ −1, we denote
A j = (−1) j+1 H ( p j , q j ),
B j = (−1) j (λ2 p j−1 p j + 2λ1 p j−1 q j + λ0 q j−1 q j ).
(6.11)
The explicit form of A j and B j for j = −1 and j = 0 is
A−1 = λ2 ,
B−1 = 0,
A0 = −(λ2 a02 + 2λ1 a0 + λ0 ),
B0 = λ2 a0 + 2λ1 .
(6.12)
Proposition 6.2 For j ≥ −1 the following identity holds
α j+1 =
B j + λ2 α
,
Aj
Proof From (6.1), we can write
α j+1 = −
p j−1 − αq j−1
( p j−1 − αq j−1 )( p j − αq j )
,
=−
p j − αq j
( p j − αq j )( p j − αq j )
then with the notation (6.11), it follows that
(6.13)