Fields Institute Communications
Volume 00, 0000

On a generalization of the radix representation - a survey
S. Akiyama
Department of Mathematics, Faculty of Science Niigata University,
Ikarashi 2-8050, Niigata 950-2181, JAPANakiyama@math.sc.niigata-u.ac.jp

T. Borbély
Department of Computer Science, University of Debrecen,
P.O. Box 12, H-4010 Debrecen, HUNGARY
tborbely@delfin.klte.hu

H. Brunotte
Haus-Endt-Strasse 88
D-40593 Düsseldorf, GERMANY
brunoth@web.de

A. Pethő
Department of Computer Science, University of Debrecen,
P.O. Box 12, H-4010 Debrecen, HUNGARY
pethoe@math.klte.hu

J. M. Thuswaldner
Department of Mathematics and Statistics, Leoben University,
Franz-Josef-Strasse 18, A-8700 Leoben, AUSTRIA
joerg.thuswaldner@unileoben.ac.at
Dedicated to Hugh C. Williams on the occasion of his 60th Birthday

1 Introductiond

Let P (X) = pd X + · · · + p0 ∈ Z[X] with p0 ≥ 2 and N = {0, 1, . . . , p0 − 1}. If
pd = 1 then P (X) is called a CNS1 -polynomial, whenever every non-zero elementof R := Z[X]/P Z[X] can be written uniquely in the form
a0 + a1 x + · · · + a` x`(1.1)

with a0 , ..., a` ∈ N , a` 6= 0; here x denotes the image of X under the canonicalepimorphism from Z[X] to R. This means that every coset Q + P Z[X] (Q ∈ Z[X])2000 Mathematics Subject Classification. 11A63.
The first author was supported by the Japanese Ministry of Education, Culture, Sports,
Science and Technology, Grant-in Aid for fundamental research 14540015, 2002–2005.
The fourth author was supported partially by the Hungarian National Foundation for Scientific Research Grant Nos. T42985 and T38225.
The fifth author was supported by project S8310 of the Austrian Science Foundation.
1 CNS is the abbreviation for Canonical Number System.
c°0000
American Mathematical Society

1

2

S. Akiyama, T. Borbély, H. Brunotte, A. Pethő, and J. M. Thuswaldner

contains a polynomial with coefficients belonging to N . The polynomial (1) will
be called the CNS-representation of the coset. The set of CNS-polynomials will bedenoted by C.
This concept was introduced by the fourth author [23] as a natural generalization of bases of canonical number systems or radix representations in algebraic
number fields, which were defined in [10] and [6]. A complex number α is the baseof a canonical number system in the algebraic number field K if and only if α is azero of an irreducible CNS-polynomial and 1, α, . . . , αd−1 is an integral basis of ZK ,where ZK denotes the maximal order of K.
In this paper we give a survey on results on canonical number systems in algebraic number fields and on CNS-polynomials. The “backward” division of polynomials, which will be defined in Section 2, plays a special rôle. Changing the bases
1, X, X 2 , . . . appropriately one obtains a mapping τP : Zd 7→ Zd , which enables
one to decide quite efficiently whether P ∈ C or not. The properties and applications of τP will be described in Sections 4 and 5. This mapping can be generalized
further and one obtains a decomposition of Rd into convex sets.
The CNS-concept was generalized to simultaneous representations of tuplesof integers in [9] and studied recently in [24]. The generalization for polynomials
over finite fields can be found in [27], where the complete characterization of CNSpolynomials over finite fields is given. Because of lack of space we will not deal withthese generalizations.
2 “Backward” division of polynomials
If pd = 1 then it is clear that every coset of Z[X]/P (X)Z[X] has an element ofdegree at most d − 1 with coefficients, which can be arbitrarily large, say
A(x) = A0 + A1 x + · · · + Ad−1 xd−1 .

(2.1)

To transform A(x) into the form (1) it is straightforward to use the following“backward” division process. Let Z0 [X] = {A(X) ∈ Z[X] : deg A < d} andTP (A) =d−1
X(Ai+1 − qpi+1 )X i ,i=0where Ad = 0 and q = [A0 /p0 ]. Then TP : Z0 [X] → Z0 [X] andA(X) = a0 + XTP (A), with a0 = A0 − qp0 .
If it causes no confusion we omit the subscript P .
Thus, to obtain the CNS representation of A(X) one has to compute the iterates
T (A), T 2 (A), . . . until T ` (A) = 0 for some ` > 0. This “backward” division process
can become divergent (e.g. A(X) = −1 for P (X) = X 2 + 4X + 2) or ultimately
periodic (e.g. A(X) = −1 for P (X) = X 2 − 2X + 2) or can terminate after finitely
many steps (e.g. A(X) = −1 for P (X) = X 2 + 2X + 2). This means that C is aproper subset of Z[X].
Let
Π(P ) = {A : TP` (A) = A for some ` > 0}
denote the set of periodic elements with respect to the mapping TP . It is clear that
we always have 0 ∈ Π(P ). Moreover P (X) ∈ C if and only if Π(P ) = {0}. Hence,
it is enough to study the map TP : Zd → Zd defined as
TP ((A0 , . . . , Ad−1 )) = (A1 − qp1 , . . . , Ad−1 − qpd−1 , −qpd ), q = [A0 /p0 ].

On a generalization of the radix representation - a survey

3

The following theorem is easy to prove.
Theorem 2.1 (Analytical conditions) If P (X) ∈ C then• all roots of P (X) are lying outside the closed unit circle, and• all real roots of P (X) are less than −1.
This theorem implies that if P ∈ C then p0 ≥ 2.
For monic P (X) ∈ Z[X] and for c > 0 let
Pc = {A(X) =d−1
X

Ai X i ∈ Z0 [X] : |Ai | ≤ c, 0 ≤ i ≤ d − 1}.

i=0

The next theorem was proved for irreducible polynomials in [20], for square-freepolynomials in [23] and in the general case in [3] and [24].
Theorem 2.2 Assume that for P (X) ∈ Z[X] the conditions of Theorem 2.1
hold. Then there exists a computable constant c > 0 such that P (X) ∈ C if andonly if every A(X) ∈ Pc has a CNS-representation.
As the set Pc is finite for all c > 0, the CNS property is algorithmically decidable. Unfortunately the constant c in Theorem 2.2 is usually large, therefore itis hard to apply it (cf. [20]). However, there are important special cases of CNSpolynomials. The first was discovered by B. Kovács [19] and proved in the generalcase in [23].
Theorem 2.3 Let P (X) = X d + pd−1 X d−1 + · · · + p0 . If p0 ≥ 2 and pd−1 ≤· · · ≤ p1 ≤ p0 and P (X) is not divisible by a cyclotomic polynomial then P (X) ∈ C.
The second special case appeared in [2] and has been generalized slightly in[26] and [3]:Pd
Theorem 2.4 Assume that p2 ≥ 0, . . . , pd−1 ≥ 0, i=1 pi ≥ 0 and p0 >Pd
i=1 |pi |, then P (X) ∈ C.
3 CNS in algebraic number fields
By the remark after Theorem 2.1 the bases of radix representations in Q correspond to the roots of X + p0 with p0 ≥ 2, i.e., they are negative integers. The
negative base representations were studied√ for the first time in [7]. The radix representations in the Gaussian integers Z[ −1] were studied by Knuth [13] (see also[14]) and by Penney [22]. In [10] all CNS in Gaussian integers were characterized.
This characterization has been extended to algebraic integers in real and imaginary
quadratic number fields in [11, 12]. The same characterization was established independently in [6]. Brunotte [4] gave a new proof without assuming irreducibilityof the quadratic polynomials.
Theorem 3.1 We have P (X) = X 2 +p1 X +p0 ∈ C if and only if −1 ≤ p1 ≤ p0
and p0 ≥ 2.
A. Kovács [15, 17] considered the possible length of periods and the size of
Π(P ) corresponding to irreducible quadratic polynomials P (X) with two complex
roots. Let θ be a root of P (X) and assume that its representation in the canonicalintegral basis 1, ω of Q(θ) is θ = a + bω, b > 0. Then he proved in [17] among others
that the cardinality of Π(P ) can only be b, b + 1 or b + 2. A full characterization of
Π(P ) for quadratic polynomials can be found in Thuswaldner [28].

4

S. Akiyama, T. Borbély, H. Brunotte, A. Pethő, and J. M. Thuswaldner

For cubic number fields much less is known.
√ Körmendi [21] described up to
one possible exception all bases of CNS in Q( 3 2) and B. Kovács and Pethő [20]in all but one totally real fields with discriminant at most 564. In the field definedby one root of the polynomial X 3 + 1749X 2 + 5975X + 5108 their result was notcomplete, because the constant appearing in Theorem 2.2 was to large. These gaps
were filled in [4]. In [1] all CNS in infinite parametric families of number rings were
established.
After some computation and proving that the conditions are necessary, Gilbert[6] proposed the following conjecture for irreducible cubic polynomials.
Conjecture 3.2 Let P = X 3 + p2 X 2 + p1 X + p0 . Then P ∈ C if and only if(i) p0 ≥ 2,(ii) p2 ≥ 0,(iii) p1 + p2 ≥ −1,(iv) p1 − p2 ≤ p0 − 2,
 p0 − 2, if p1 ≤ 0,
p0 − 1, if 1 ≤ p1 ≤ p0 − 1,(v) p2 ≤p0 ,if p1 ≥ p0 .
We will come back later to this conjecture, but will mention already here thatthe situation is much more complicated if p0 ≥ 6.
For higher degree fields nearly nothing is known. The only general result is due
to B. Kovács [19].
Theorem 3.3 There exists in ZK a CNS if and only if ZK is monogenic, i.e.,there exists an α ∈ ZK such that {1, α, . . . , αd−1 } is an integral basis in ZK .
Combining this theorem with a result of Győry [8] we obtain that up to translation by integers there exist only finitely many CNS in ZK .
4 Brunotte’s mapping
As we mentioned before, Theorem 2.2 is not efficient enough to decide the CNS
property of a polynomial. Brunotte [4] observed that the basis transformation{1, x, . . . , xd−1 } →wj={w1 , . . . , wd },dXpi xi−j , j = 1, . . . , di=jof R implies a nicer and much better applicable transformation as TP is. Indeed, if
A(x) =dX

Āj wj ,

then

j=1

TP (A) =−tw1 +dX
j=2·
Āj−1 wj , where t =¸
p1 Ā1 + · · · + pd Ād.
p0

Hence TP implies the mapping τP : Zd → Zdµ ·¸¶
p1 A1 + · · · + pd Ad
τP (A) = −, A1 , . . . , Ad−1 ,p0where A = (A1 , . . . , Ad ). The mapping τP will be called Brunotte’s mapping.
Scheicher and Thuswaldner [26] made the same discovery independently.

On a generalization of the radix representation - a survey

5

Brunotte’s mapping is easy to implement and it is immediately clear that P ∈/Cif either the analytical conditions of Theorem 2.1 do not hold or there exists 0 6= A ∈Zd and ` > 0 such that τP` (A) = A. Its importance relies on the following theorem,
which is in some sense the converse of the last statement and makes it possible todecide the CNS property. Moreover it enables a far reaching generalization. It wasproved originally in [4] and refined in [3]. The present version was published in [1].
Theorem 4.1 Suppose that E ∈ Zd has the following properties:• (1, 0, . . . , 0) ∈ E,
• −E ⊆ E,• τP (E) ⊆ E,
• for each e ∈ E there exists some ` > 0 with τP` (e) = 0.
Then P (X) ∈ C.
Such a set E will be called the set of witnesses of P ∈ C.
5 Applications of Brunotte’s mapping
Applying Theorem 4.1 Brunotte was able to characterize all CNS trinomials[5].
Theorem 5.1 If d > 2 then the following assertions hold:(i) X d + bX + c belongs to C if and only if −1 ≤ b ≤ c − 2,(ii) if 1 < q < d and q 6 |d then X d + bX q + c ∈ C if and only if 0 ≤ b ≤ c − 2.
Akiyama et al. [1] examined Gilbert’s Conjecture 3.2 and proved it in somecases, e.g. if• p1 ≤ −1, p2 ≤ p0 − 2 and −1 ≤ p1 + p2 ≤ 0,
• p1 ≤ −1, 0 ≤ p2 < min{p0 − 1, 2p0 /3} and 1 + p1 + p2 ≥ 0,
• ≤ p1 ≤ p0 − 1 and 0 ≤ p2 ≤ (2p0 − 1)/3.
On the other hand they found that Gilbert’s Conjecture does not hold if p0 ≥ 6.
We present some counterexamples:(i) p1 ≤ 0. Let 2 ≤ p1 + p2 ≤ −p1 and p0 ≤ min{p2 − p1 , p1 + 2p2 + 1} then(1, −1, −1) is a periodic element whose period is (1, −1, −1); 2, 1, −1, −1.
Here and in the sequel we present a period by giving a vector and the firstcoordinate of the following vectors.
2(ii) 1 ≤ p1 ≤ p0 − 1. Let p0 ≥ 28 and 7p0 −5p+ 1 ≤ p1 ≤ −p0 + 32 p2 . Then the6element (1, −3, 1) is periodic with period (1, −3, 1); 3, −2, −2, 3, 1, −3.
(iii) p1 > p0 . Let p0 + 12 p2 + 1 ≤ p1 < p0 + 23 p2 − 13 . Then the element (3, −2, 1)
is periodic with period (3, −2, 1); −2, 1, 1, −2.
We visualize the situation with the example p0 = 474. In this case there are396, 830 CNS-polynomials and 52, 046 polynomials that violate Gilbert’s conjecture.
The point (p1 , p2 ) on Picture 1 corresponds to the polynomial X 3 +p2 X 2 +p1 X +p0 .
The displayed region is defined by the inequalities from Conjecture 3.2. The graypoints correspond to members of C and the black ones to those, which violate
Gilbert’s conjecture. From this picture it is to be expected that the set of cubic
CNS polynomials has a complicated structure.

6

S. Akiyama, T. Borbély, H. Brunotte, A. Pethő, and J. M. Thuswaldner

Picture 1. CNS polynomials for p0 = 474.
Theorem 2.1 implies that for fixed degree and given p0 ≥ 2 there exist only
finitely many CNS-polynomials. Especially interesting is the case p0 = 2, i.e. thegeneralizations of the binary expansion. Using Brunotte’s algorithm A. Kovács[18] computed all binary CNS polynomials of degree d ≤ 8. The result of his
computation is displayed in the next table.
Degree1
Number of CNS-polynomials 1

2
3

3
4

4
12

5 6
7 25

7
12

8
20

To show how hard it is to decide whether a polynomial belongs to C we give
two examples: For X 8 + 2X 7 + 3X 6 + 3X 5 + 3X 4 + 3X 3 + 3X 2 + 3X + 2 thesmallest set of witnesses has 241, 719 elements, while for X 3 + 317X 2 + 632X + 317
has 1, 308, 322 elements.
A natural question is whether the CNS-property belongs to the NP or to the
coNP class? The above examples indicate that the CNS-property cannot be decided in polynomial time. In the other direction Scheicher and Thuswaldner [26]noticed that if P (X) = X 3 + 196X 2 + 341X + 199 then the length of the periodof (−11, 10, −6) is 84. They conjecture that already cubic polynomials can havearbitrarily long cycles. Note that (3, −2, 1) is another periodic point of τP , butwith period length 7.
6 Generalization of Brunotte’s mapping
In an earlier stage of our investigations we tried to prove algebraic propertiesof the set C. It turned out that C is not closed under addition, multiplication andincrementation by 1. However, some of these algebraic properties are valid for large
subsets of C. Especially, the examples (e.g. x3 + 80x2 + 117x + 89) where P (x) ∈ C
but P (x) + 1 6∈ C seem to be rather exceptional.
There are of course trivial algebraic results (which do not show anything new)
if one appropriately restricts to subsets of C. Let for example M be the set of CNS
polynomials of degree 2 or of degree 3 which satisfy the assumptions of Proposition
3.3 in [1]. Then, if Q = X + k, k ≥ 2 and P ∈ M then P + Q ∈ C.
The only non-trivial algebraic result was proved in [5]. It asserts that if P (X) ∈C and k ≥ 1 then P (X k ) ∈ C.
A closer look at C showed that it (or a related set) has to be the union of convex
bodies. To show this property we followed Paul Erdős instruction: “If you cannot
solve a problem, then try to generalize it and solve the more general problem.”Brunotte’s mapping allows such a generalization.

On a generalization of the radix representation - a survey

7

Let r = (r1 , . . . , rd ) ∈ Rd , rd 6= 0. With r we associate the mapping τr : Zd →Z by the following way: if a = (a1 , . . . , ad ) ∈ Zd then letdτr (a) = (−[ra], a1 , . . . , ad−1 ),where ra = r1 a1 + · · · + rd ad . Obviously this is a generalization of Brunotte’s
mapping by taking r = ( pp10 , . . . , ppd0 ).
Let
Cd = {r : for all a ∈ Zd there exists ` > 0 such that τr` (a) = 0}.
The next theorem shows that the set of mappings τr has some convexity property.
Theorem 6.1 Let r1 , . . . , rk ∈ Rd and a ∈ Zd be such that τr1 (a) = · · · =τrk (a). Let s be any convex linear combination of r1 , . . . , rk . Then we have τs (a) =τr1 (a) = · · · = τrk (a).
This theorem implies immediately the following corollary
Corollary 6.2 Let r1 , . . . , rk ∈ Rd have the same period, i.e. τr`1 (a) = · · · =τr`k (a), ` = 0, . . . , v and a = τrv1 (a). Then if s lies in the convex hull of r1 , . . . , rk
the mapping τs is periodic and has the same period as τr1 .
¢¡253
example,it is¡ easy to ¢check that for the plane vectors r1 = 381
254 , 254 , r2 =¡ 421For
¢253
344 176
254 , 254 and r3 = 254 , 254 the corresponding mappings have the same period(−2, 1); 3, −2, 1, 1, −2, hence, the corresponding mapping for any point lying in thetriangle r1 , r2 , r3 has this period too.
¡¢
¡ 421 253 1 ¢253
1A¡ three-dimensional
example is: r1 = 382
254 , 254 , 254 , r2 = 254 , 254 , 254 and¢176
1
r3 = 344
254 , 254 , 254 . Here is the period (3, −2, 1); −2, 1, 1, −2, 3.
Theorem 4.1 can be generalized for this setting.
Theorem 6.3 Let r1 , . . . , rk ∈ Rd and denote by H the convex hull of r1 , . . . , rk .
For z ∈ Zd take m(z) = min1≤i≤k {−[ri z]} and M (z) = max1≤i≤k {−[ri z]}. Suppose that there exists a finite set E, which satisfies the following conditions:• ±e ∈ E for all d-dimensional unit vectors e,• for each z = (z1 , . . . , zd ) ∈ E andj ∈ [min{m(z), −M (−z)}, max{−m(−z), M (z)}] ∩ Z
we have (j, z1 , . . . , zd−1 ) ∈ E,j
• ∩∞j=1 τri (E) = {0} for each i ∈ {1, . . . , k}.
Then H ⊆ Cd .
¡
¢ ¡
¢ ¡
¢ ¡
¢
For example the square with vertices 13 , 13 , 13 , 23 , 23 , 13 , 32 , 13 is a subset
of C2 as one can show by using the set of witnesses E = E1 ∪ (−E1 ) ∪ {(0, 0)},where E1 = {(1, 0), (0, 1), (1, −1), (1, 1), (2, −1), (1, −2), (2, 0), (0, 2), (1, 2), (2, −2)}.
It is easy to see that C2 is a subset of the region
R2 = {(γ1 , γ2 ) : −1 ≤ γ1 < 2, 0 ≤ γ2 < 1, −γ1 ≤ γ2 < γ1 + 1}.
On Pictures 2³and 3 ´we present two approximations of C2 . We displayed thereall (γ1 , γ2 ) = pp01 , pp20 ∈ R2 with p0 , p1 , p2 ∈ Z. For Picture 2 we have chosen
p0 = 60, and for Picture 3 we took p0 = 174. The light-gray points belong and thedark-gray points do not belong to C2 . The status of the points lying on the blacklines could not be decided for the chosen precision. However, it can be shown thata considerable part of the black points does indeed belong to C2 .

8

S. Akiyama, T. Borbély, H. Brunotte, A. Pethő, and J. M. Thuswaldner

Picture 2. An approximation of C2 , p0 = 60.

Picture 3. Better approximation of C2 , p0 = 174.
The top boundary of Pictures 1 and 3 seems to be very similar. Unfortunately
we do not understand yet the relation between the two sets. By the last theorem
Cd is the union of convex sets, but it is not clear whether finite or countably many
sets appear in this union.
References[1] S. Akiyama, H. Brunotte and A. Pethő, Cubic CNS polynomials, notes on a conjectureof W.J. Gilbert, J. Math. Anal. and Appl., to appear[2] S. Akiyama and A. Pethő, On canonical number systems, Theoret. Comput. Sci., 270(2002), 921 - 933.
[3] S. Akiyama and H. Rao, New criteria for canonical number systems, preprint[4] H. Brunotte, On trinomial bases of radix representations of algebraic integers, Acta Sci.
Math. (Szeged), 67 (2001), 407-413.
[5] H. Brunotte, Characterization of CNS trinomials, Acta Sci. Math. (Szeged), 68 (2002),
673-679.
[6] W.J. Gilbert, Radix representations of quadratic fields, J. Math. Anal. Appl., 83 (1981),
264-274.
[7] V. Grünwald, Intorno all’aritmetica dei sistemi numerici a base negativa con particolare riguardo al sistema numerico a base negativo-decimale per lo studio delle sue analogie coll’aritmetica ordinaria (decimale), Giornale di matematiche di Battaglini, 23 (1885),203-221, 367.
[8] K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné, III, Publ.
Math. Debrecen, 23 (1976), 141-165.
[9] K.-H. Indlekofer, I. Kátai and P. Racskó, Number systems and fractal geometry, Probability theory and applications, Essays to the Mem. of J. Mogyoródi, Math. Appl., 80 (1992),
319 -334.
[10] I. Kátai and J. Szabó, Canonical number systems for complex integers, Acta Sci. Math.
(Szeged), 37 (1975), 255-260.
[11] I. Kátai and B. Kovács, Kanonische Zahlensysteme in der Theorie der quadratischen algebraischen Zahlen, Acta Sci. Math. (Szeged), 42 (1980), 99-107.

On a generalization of the radix representation - a survey

9[12] I. Kátai and B. Kovács, Canonical number systems in imaginary quadratic fields, Acta
Math. Acad. Sci. Hungar., 37 (1981), 159-164.
[13] D. E. Knuth An imaginary number system, Comm. ACM, 3 (1960), 245-247.
[14] D. E. Knuth The Art of Computer Programming, Vol. 2 Semi-numerical Algorithms, Addison Wesley (1998) London 3rd-edition.
[15] A. Kovács, On expansions of Gaussian integers with non-negative digits, Math. Pannon.,
10 (1999), 177-191.
[16] A. Kovács, On the computation of attractors for invertible expanding linear operators in
Zk , Publ. Math. Debrecen, 56 (2000), 97-120.
[17] A. Kovács, On canonical expansions of integers in imaginary quadratic fields, Acta Math.
Hung., 93 (2001), 347-357.
[18] A. Kovács, Generalized binary number systems, Ann. Univ. Sci. Budap. Rolando Etvös,
Sect. Comput., 20 (2001), 195-206.
[19] B. Kovács, Canonical number systems in algebraic number fields, Acta Math. Acad. Sci.
Hungar., 37 (1981), 405-407.
[20] B. Kovács and A. Pethő, Number systems in integral domains, especially in orders ofalgebraic number fields, Acta Sci. Math. (Szeged),√55 (1991) 287-299.
[21] S. Körmendi, Canonical number systems in Q( 3 2), Acta Sci. Math. (Szeged), 50 (1986),
351 - 357.
[22] W. Penney, A ’binary’ system for complex numbers, J. Assoc. Comput. Mach., 12 (1965),247-248.
[23] A. Pethő, On a polynomial transformation and its application to the construction of apublic key cryptosystem, Computational Number Theory, Proc., Walter de Gruyter Publ.
Comp. Eds.: A. Pethő, M. Pohst, H.G. Zimmer and H.C. Williams, 1991, pp 31-43.
[24] A. Pethő Notes on CNS polynomials and integral interpolation, to appear.
[25] K. Scheicher, Kanonische Ziffernsysteme und Automaten, Grazer Math. Ber., 333 (1997),
1-17.
[26] K. Scheicher and J.M. Thuswaldner, On the characterization of Canonical Number Systems, Osaka J. Math., to appear.
[27] K. Scheicher and J.M. Thuswaldner, Digit systems in polynomial rings over finite fields,
Finite Fields Appl., to appear.
[28] J.M. Thuswaldner, Attractors for invertible expanding linear operators and number systems
in Z2 , Publ. Math. Debrecen, 58 (2001), 423-440.