Periodic Elements and Number Systems in Q(a) - Science Direct

MATHEMATICAL ~%~UTER MODELLING PERGAMON

Mathematical

and

Computer

Modelling

38 (2003)

783-788 www.elsevier.com/‘l~-)cate/mcm

Periodic Elements and Number Systems in Q(a) G.

Department P&mbny

FARKAS of Computer Algebra, E&v& Lorbnd University Pdter s&&ny l/C, H-1117 Budapest, Hungary farkasgQcompalg.inf.elte.hu

Abstract-Let

us consider an arbitrary quadratic extension of the field of rational numbers Our prospective purpose is to give for an arbitrary algebraic integer a, if any, such a digit set. that constitutes a number system with cr. In this paper, we deal with the periodic elements of systems given in Q(a) and prove that either the modulus of them or that of their conjugate is less than 1. On the basis of this result, we hope that there exists some algorithm which provides number systems. @ 2003 Elsevier Ltd. All rights reserved.

Keywords-Generalized

number system, Periodic element.

1. INTRODUCTION 1.1.

Number

Systems

in Quadratic

Extension

Fields

We call the complex number J3 an algebraic number, if ,0 is a root of some polynomial 0 # f(x) 6 a4 It is a well-known fact in number theory that for every algebraic number ,L3,there uniquely exists this f(z), so that f( IC) is an irreducible manic polynomial. Then, we say that f(z) is the minimal polynomial of p, each root of f( x ) is a conjugate of /3, and ,B is an algebraic integer if all the coefficients of f(x) are integers. /3 is a quadratic algebraic number if deg(f(z)) = 2. Let us consider now number fields T and F, and assume that F is a subfield of T: i.e., T is an extension of F. We claim that T is an algebraic extension of F if each element of T is a root of some nonzero polynomial f(z) E F[s]. Let F be a subfield of T and A 2 T. Then we denote the smallest subset of T which contains A and F with F(A). If A’s only element is p, i.e., A = {a}, we normally write F(P) instead of F(W). In this paper, we deal with the mathematical structures type Q(p), called quadratzc extension. fields, where ,f? is a quadratic algebraic number, and Q is the set of the rational numbers. It can be proved that every quadratic extension field takes on type Q(a), where D is a square-free integer, and IDI > 1. We call Q(a) an imaginary quadratic extension field if D < 0. whereas we call it a real quadratic extension field, provided D > 0. Let I be the set of algebraic integers in Q(a). For some p = c + cl& E {Q(a); let p = c - da be the algebraic conjugate of ,B and r(p) = c the rational, i(/3) = d the irrational part of /3, where c, d E Q. 0895-7177/03/$ - see front doi: lO.l016/SO895-‘7177(03)00279-6

matter

@ 2003

Elsevier

Ltd.

All

rights

reserved.

Typeset

by &S-TJ$

784

G. FARKAS

1. Let a E I and E,(C I) be a complete residue system mod cy containing 0, i.e., such a collection of fo = 0, fi, . . . , ft-1 E I for which for every y E I there exists a unique f E E, such that DEFINITION

Y = @Yl + f,

(1)

with a suitable y1 E I. Then, we can say that (a, Ea) is a coefficient system. from now on, we call the elements of E, digits or coefficients and (Y is the base number of (a, E,). DEFINITION

2. We say that (CX,E,)

is a number system in I if each y E I can be written as a

finite sum y = eo + ela + . . 9+ ekak,

(2)

where e, E E,, i = 0, 1, . . , k. Naturally in this case (Yis the base of the number system (a, E,). The uniqueness of representation (2) follows from the fact that E, is a complete residue system mod CY. Before examining an arbitrary (a, E,), we have to introduce certain concepts. DEFINITION

3. Let the J : I -

I function

be defined in the following

holds, then J(y) = 71. Then, we can speak about transition iterate of J.

y 2

way: if equation

(1)

71. J” denotes the k-fold

4. An arbitrary K E I is a periodic element if there exists a positive integer k such that J’“(n) = x. Let P be the set of periodic elements, and G(P) the directed graph to be obtained by directing an edge from r to J(r) for every r E P.

DEFINITION

Our long-term objective is to give for an arbitrary (YE I, if any, such an E, digit set that (cu,E,) will constitute a number system. The investigation of graph G(P) and set P is indispensable in finding the appropriate digit set, because, as we will see later, (a, E,) is a NS if and only if P = (0). In this paper, we focus our attention on Q(A) an d we render the upper bound of the modulus of elements of P to an arbitrary cr E I. 1.2. Previous

Results

in This Field

The research in this field has two main directions. (*) For a given cy find such a digit system E,, if any, for which (Q, E,) is a number system. (**) For a given LYand digit set EC = {O,l,. . . , /N(cv)) - 1) decide whether EC is a complete residue system moda and (a, EC) is a number system or not. Problem (**) was solved for quadratic extension fields in [l-4]. It was shown that only very special numbers cycan be served as bases with such special digit set. In [5,6], the number, location, and structural properties of periodic elements were fully described in imaginary quadratic fields. With respect to problem (*), Steidl observed in [7] that in the case of Gaussian integers, a good strategy for the choice of an appropriate digit set is to take one for which maxfEE- IfI is close to the minimum. Later KAtai [8] proved that if I is the set of integers in some imaginary quadratic extension field, then cy E I is a base of a number system with an appropriate digit set if and only if (Y# 0, Ial # 1, 11-al # 1. Kovdcs proved in [9] that for a given expansive M E Rkxk providing that the operator norm JIM-1j(2 < l/(1 + &), th ere is always a digit set E such that (M, E) is a number system. It has been shown in [lo] that if Ial 2 2, Iii1 2 2, then there always exists an E, for which (a, E,) is a number system. The digit set was explicitly computed. The same result was obtained in [9] using a different digit set construction. It has been proved in [ll] that we can find an E, In detail: for an arbitrary integer coefficient system for which G(P) has a simple structure. LY E Q(a), where Q,Q - 1 is not a unit, 1 < min(lal, I&() < 2, and D > 1 is a square-free integer, there exists such an E, coefficient system for which G(P) is the disjoint union of loops, 0 it contains if either IcyI > 2 > 161, d > 0, or @J > 2 > la], Q > 0, and beside the loop 0 -

785

Periodic Elements and Number Systems

only circles of order two of type r (-r) --+ 7~~if either (CYI> 2 > J&l, si: < 0 or 161 > 2 > ICY/, cx < 0 holds. In [12], we made some statements in connection with the location and number of periodic elements in Q(a), which we will discuss in detail after we have introduced the relevant definitions in Section 2.

2. THE PRESENTATION

OF OUR ASSERTION

2.1. Observations One can easily observe that if ~1:E 1 is a base of a number system in Q!(a) digit set, then the following assertions are valid:

with a suitable

(1) Q # 0, (2) a # unit, (3) 1 - a: # unit, (4) (5) (6) (7)

Ial > 1, ISI > 1, if IcyI, 1~1 > 1, then for each y E I the path y, J(y), J2(r), G(P) is a disjoint union of directed circles, (a, E,) is a number system if and only if P = (0).

. is ultimately

periodic,

Assertions (1) and (2) are obvious. Assume that 1 - (Y = E = unit and (a, E,) is a number system. Let f E E,, f # 0, y = fs6, where 6 = EC Then, y = f + cry and y f 0, and consequently y cannot be expanded as (2). Assume that ICEI< 1 and (0, E,) is a number system. Then, the set of y having the finite representation (2) is bounded, while the whole set I is not bounded, which is a contradiction. Let us observe finally that, (a, E,) is a number system if and only if (6: EN) is a number system, where l?, consists of the set of the algebraic conjugates of the elements of E,. This implies that (61 > 1 is also necessary. 2.2. K-Type

Digit

Sets

Let us consider now Q(a). It is known that { 1, fi} is an integral basis in I. Let E = H. b = fl and for some cy E Q(A) and let d = N(Q) = a&. Let 0 = u + b& and Eg’6) be the sets of those f = k + 1fi, k, 1 E II! for which f& = (Ic + 1 &)(a - b a) = (ka - b/2) + (la - kb) fi = r + s fi satisfy the following conditions: - if (E, 6) = (1, l), then r, s E (-ld(/2,ldl/2]; - if (.E,6) = (-1, -l), then T, s E [-ldl/2,ld(/2); if (E, b) = (-1, l), then T E [-ld(/2,ldl/2), s E [-ldl/2,ldl/2); - if (E, S) = (1; -l), then T E (-ld1/2,ldl/2], s E (-ldl/2, (d(/2). It is known from number theory that Et”) is a complete residue system mod Q. F’rom now on, we call the above constructed coefficient sets K-type digit sets. 2.3. The Formulation

of Our Theorem

THEOREM 1. Let cu = a C b fi be an arbitrary algebraic integer in Q(a), for which CB# 0, Q$(Y* 1 is not a unit, 1 < min(lcu(, 161) < 2 are valid. Let, in addition, E, be an arbitrary K-type digit set. Then, for each 7r E P.

ITI < 1, IFI < 1, 2.4. Preparing

if sgn (a) = sgn (b),

and

if sgn (cz) # sgn (b).

the Proof

R.EMARK 1. Since during the proof we never specify the value of (E, S), therefore our proof will hold true for each Et”). Observe that if d is an odd number, then we get the same digit set for arbitrary value of (E, 6). If d is an even number addition we can say that

786

G. FARKAS j-J?+? a

=

EL-E,-@,

because f (-a)

Eh6) CY = E$d’,

= --T - s A,

and

because fo = r - s A.

Thus, if we have proved our assertions in case (a] > ]E], we get the theorem in case ((u] < 161 simply by reversing the roles of ]a(, ]c%]and those of Eg’6’, Et’-6’. Further, we assume that IQ] > 2 and 1 < ]&I < 2 and an arbitrary element of the set {Ei”l), EL”-‘), EA-lll), Ei-l’-l)} will be denoted by E,. REMARK 2. In [ll], we dealt with the case Icy], ]&I > 2 and now we observe that if ]a], ]~r] < 2, then cy = -6 = fi and fi - 1 is a unit, and therefore, we do not need to investigate these two cases. Thus, we further assume that min(]a(, ]cu]) < 2 and max((cr(, I&]) > 2. REMARK 3. Paper [ll] also shows that for each n = (p + q 1/2) E P \ implies that sgn (p) # sgn (q); otherwise ]7r] 2 & + 1 would be valid. p > 0 and p < 0 are the same, we can avoid the proof of case p > assume that for arbitrary 7r = (p + q &) E P \ (0)~ < 0 holds. Then, q > 0 and ?i < 0 are always maintained.

{O}]r] < fi is true. This Since the proofs of cases 0 and in what follows we naturally, the inequalities

REMARK 4. We would point out that in [ll] we gave the number systems in the case of IQ], ]&I 5 6 + 3 fi for each possible cy E I. It can be easily checked that our theorem obviously holds for the values a, 6, and thus, in the following we can assume, without losing generality, that JCX]>6+3fi. DEFINITION 5. z = IdI/

- max(]a],]b]

fi).

The number z plays an important role in the estimating of the modulus of the periodic elements. Let us now examine the connection between 5 and &. ]a] = ]a] + ]b( fi = Id]/2 + Id]/2 - 2z - I&(, from which Jd] = ]fi]]~y] = ]~](]d] - 2x - Ia]) = ]~]]d] - 2z]h] - ]612, and thus,

x = (I4 - 1)I4 - Id2 216)

3. THE 3.1. The Number

PROOF

and Position

OF

.

OUR

of the Periodic

THEOREM

Elements

This section includes statements and their corollaries which we use in the proof of our theorem. The following lemmas describe the exact location and the number of the periodic elements. Since the proofs are included in [12], we do not give them in this paper. LEMMA 1. (a(,Jbj fi

< ]d]/2.

COROLLARY 1. fl E E,. Let us assume that f = 1. from this we get that fCi = d = a-b 4 = rfsfi. Thus, jr/ = (al < (d(/2, and (sl = IbJ < Jd[/2. W e can obtain the same result for f = -1. LEMMA 2. Let us assume that )&I 2 (a+

1)/2. Then for each 7r E P holds 1~1 < 1.

COROLLARY 2. It is obvious that in case (&I 2 (fi+ can assume from now on that

1)/2, our theorem is proved, and thus, we

COROLLARY 3. Weget

x 0. The proofs of the other cases can be completed in a similar way, and therefore, we do not present them here, and from now on, we assume that cr,& > 0. In addition, let us assume indirectly that there exists such a or which [7r\ > 1. Then we have to investigate the following three cases. ?r=p+q2/IsEP\{O}f

3.2.1.

The case x = fl

According to the assertion of Corollary 2, we can see that in this case 7r E E,. because 7r = Oo + n; i.e., 7r 5 0 is valid. Thus, we have got In/ # 1.

3.2.2.

Then r $ P,

The case 7~< -1

Let us assume that n < 0. We can write: f = ~(1 - o) > 0, f = ?i(l - &) > 0, from which we get that f& = r + s& > 0, fo = r - sfi > 0. Thus, we can see that T > 0 and /r/ > !s\v'%!. There are two possible cases. If s < 0, then If&l = Inl(ldl - I&l) = Ir1 - (s( fi < (d[/2, and from this we get t,hat (~1
0, then I~cx/ = liil(ldl-- Ial) = (?i((2z+l&[) = (r-1- Is{&, and thus, (s[ ~‘2 = ir; -- /ii((22iI&\). It is known that If&l = 17rl(ldl - l&l) = 17-l+ (s( 4 = 2!rl - (iil(2z + j&l), and therefore, we get for 7r that. InI < k-4 - 14 (22 + PI) < 1

I4 - 14

’

since lii( = jpl + \ql fi 2 1 + 4. It is obvious that the case YT< -1 never appears because we have found that 7r < 0 implies 1~1 < 1. We have completed the proof of our theorem in this case.

3.2.3.

The case n > 1

Let us assume now that ?r > 1. Then we know from Lemma 4 that there exists a 7r’ E P and 7r’ = n - 1. Thus, the following assertions hold: f = rr(1 - o) < O! f = F(! .- fi) > 0. f’ = ~‘(1 - a’) < 0, f’ = ?i’( 1 - 6’) > 0; i.e., fir! = r + s fi7r < 0, fo = T --- $9v% > 0, f’~ = r’ + s’ fi < 0, f”o = T’ - s’ fi > 0. We get that s, s’ < 0 and b-1 < js\ ~‘2. In this case, 71’- 71= n = -1, and therefore, Lemma 3 implies that s’ = s -b; i.e., (~‘1 = (6/$ is/. We get that,

(4) Now let us estimate the value of T. We can write the relation

If4 = 14WI - 14) = 14(22+ Ial) > I4 - Id 1

(5)

because x > 1 is valid. Since Jr-1< IsI fi holds, either If61 = IsI fi - I’rj or If&l =; is/ & + /rl is true. The first case is impossible, because If&l = [sl fi - Ir[ 5 (ld(/2) ~“2 would be valid

G. FARKAS

788

and then 7r < 1, which is a contradiction. Thus, we get that If&[ = [s( fi + Irl. Then it comes from (5) that 1~1> IdI - IsI &‘- Jc~(.Equation (4) implies that jr1 > IdI - /LY/- (ld(/2) Jz+lbl &, and thus, 17-1> Jdl - I&) - (ld1/2) fi+ (ld1/2) - 5 - /&I, and from this we get that

Id

’

345 --y-

Id( -

z -

2 (6(.

Using Corollary 3, i.e., 5 < [dl/2(Jz+ 1)2 and the facts Irl < IdI/ and Id( > l&1(6 + 3fi), we can compute easily that (6) never holds. We have arrived at a contradiction in all cases; thus, 1~) < 1. The proof is completed. I

REFERENCES 1. I. K&tai and J. Szab6, Canonical number systems for complex integers, Acta Sci. Math. 37, 255-260, (1975). 2. I. KBtai and B. Kovics, Kanonische ‘Zahlensysteme bei reelen quadratischen algebraischen Zahlen, Acta Sci. Math. 42, 99-107, (1980). 3. I. K&tai and B. Kovhcs, Canonical number systems in imaginary quadratic fields, Acta Math. Hung. 37, 159-164, (1981). 4. W. Gilbert, Radix representations of quadratic fields, J. Math. Anal. and Appl. 83, 264-274, (1981). 5. A. Kovics, On expansions of Gaussian integers with non-negative digits, Mathematics Pannonica. 10 (2), 177-191, (1999). 6. A. Kov&s, Canonical expansions of integers in imaginary quadratic fields (to appear). 7. G. Steidl, On symmetric representation of Gaussian integers, BIT 29, 563-571, (1989). 8. I. K&a& Number systems in imaginary quadratic fields, Annales Univ. Sci. Budapest, Sect. Comp. 14, 159-164, (1994). Conference on Applied 9. A. KovBcs, On number expansion in lattices, In Proceedings of the 5 th International Infomatics, Eger, Hungary (to appear). 10. G. Farkas, Number systems in real quadratic fields, Annales Univ. Sci. Budapest, Sect. Comp. 18, 47-60, (1999). 11. G. Farkss, Digital expansion in real algebraic quadratic fields, Mathematicu Pannonica. 10 (2), 235-248, (1999). 12. G. Farkas, Location and number of periodic elements in Q, Annales Univ. Sci. Budapest, Sect. Comp. 20, 133-146, (2001).