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12 Conversion Algorithm of Matrix Decomposition in Ordinary Continued Fraction

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

1. Aleksandrov, A. G.: Computer investigation of continued fractions. In: Algorithmic studies in

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)

3. Bruno, A.D.: Continued fraction expansion of algebraic numbers. Zh. Vychisl. Mat. Mat. Fiz.

4, 211–221 (1964)

4. Bruno, A.D.: Universal generalization of the continued fraction algorithm. Cheb. Sb. 16(2),

35–65 (2015)

5. Dobrovol’skii, N.M.: Hyperbolic Zeta function lattices. Available from VINITI, Moscow, No

6090–84 (1984)

6. Dobrovol’skii, N.M.: Quadrature formulas for classes Esα (c) and Hsα (c). Available from

VINITI, Moscow, No 6091–84 (1984)

7. Dobrovol’skii, N.M.: About the modern problems of the theory of hyperbolic zeta-functions

of lattices. Cheb. Sb. 16(1), 176–190 (2015)

8. Dobrovol’skii, N.M., Dobrovol’skii, N.N.: About minimal polynomial residual fractions for

algebraic irrationalities. Cheb. Sb. 16(3), 147–182 (2015)

9. Dobrovol’skii, N.M., Yushina, E.I.: On the reduction of algebraic irrationalities. In: Algebra

and Applications: Proceedings of the International Conference on Algebra, Dedicated to the

100th Anniversary of L. A. Kaloujnine, Nalchik, KBSU, pp. 44–46 (2014)

10. Dobrovol’skii, N.M., Dobrovol’skii, N.N., Yushina, E.I.: On a matrix form of a theorem of

Galois on purely periodic continued fractions. Cheb. Sb. 13(3), 47–52 (2012)

11. Dobrovol’skii, N.M., Sobolev, D.K., Soboleva, V.N.: On the matrix decomposition of a reduced

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)

13. Frolov, K.K.: Upper bounds for the errors of quadrature formulae on classes of functions.

Dokl. Akad. Nauk SSSR. 231(4), 818–821 (1976)

14. Frolov, K.K.: Quadrature formula on the classes of the functions. Ph.D. thesis. Computer

Centre of the Academy of Sciences of USSR (1979)

15. Podsypanin, V.D.: On the expansion of irrationalities of the fourth degree in the continued

fraction. Cheb. Sb. 8(3), 43–46 (2007)

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)



140



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17. Podsypanin, E.V.: On the expansion of irrationalities of higher degrees in the generalized

continued fraction (Materials V. D. Podsypanin) the manuscript of 1970. Cheb. Sb. 8(3), 47—

49 (2007)

18. Prasolov, V.V.: Polynomials. Algorithms and Computation in Mathematics Algorithms and

Computation in Mathematics, vol. 11. Springer, Berlin (2004)

19. Roth, K.F.: Rational approximations to algebraic numbers. Mathematika (1955). doi:10.1112/

S0025579300000644

20. Trikolich,

E.V., Yushina, E.I.: Continued fractions for quadratic irrationalities from the field



Q( 5). Cheb. Sb. 10(1), 77–94 (2009)

21. Weyl, H.: Algebraic Theory of Numbers. Princeton University Press, Princeton (1940)

22. Yushina, E.I.: About some the reduction of algebraic irrationalities. In: Modern Problems of

Mathematics, Mechanics, Computer Science: Proceedings of the Regional Scientific Student

Conference, Tula, pp. 66–72 (2015)

23. Yushina, E.I.: On some generalized Pisot number. In: University of the XXI Century: Research

within Academic Schools: Proceedings of the Russian conference, Tula, pp. 161–170 (2015)



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−



)

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)



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