ON SOLVING RELATIVE NORM EQUATIONS IN ALGEBRAIC NUMBER FIELDS

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1 MATHEMATICS OF COMPUTATION Voume 66, Number 27, January 997, Pages S (97) ON SOLVING RELATIVE NORM EQUATIONS IN ALGEBRAIC NUMBER FIELDS C. FIEKER, A. JURK, AND M. POHST Abstract. Let Q E Fbeagebraic number fieds and M Fafree o E -modue. We prove a theorem which enabes us to determine whether a given reative norm equation of the form N F/E (η) = θ has any soutions η M at a and, if so, to compute a compete set of nonassociate soutions. Finay we formuate an agorithm using this theorem, consider its agebraic compexity and give some exampes.. Introduction Soving norm equations is a centra probem in the area of agebraic number theory. Athough there is an agorithm for soving absoute norm equations (e.g. see [] or [8, 5.3, 6.4]), none (except the absoute one) exists in the reative case. We outine a new agorithm to decide whether a reative norm equation has soutions at a and then, if there are soutions to compute a compete set of nonassociate (with respect to units of reative norm ) soutions. Finay we discuss the compexity of this agorithm and give some exampes. 2. Preiminaries First, we introduce severa definitions and notations. We consider the foowing situation: F = E(β) n E = Q(α) m Q Let α = α (),..., α (m) be the roots of the monic irreducibe poynomia f Q[t], E := Q(α). Furthermore, et β be a root of a monic irreducibe poynomia g E[t]and F:= E(β). We assume that α (),..., α (m) denote the rea roots of f and that α (m+) = α (m+m2+),...,α (m+m2) =α (m+2m2) are in C\R. For an arbitrary η E we define η (i) ( i m) as the image of η under the Q-isomorphism from E to E (i) := Q(α (i) )which maps α to α (i). The norm of an eement η of E is defined in the usua way: N(η) :=N E/Q (η):= m i= η(i). The definitions for F are essentiay the same, but here we have to be carefu about which fied we are using as base fied. In genera, the conjugates cannot be ordered in rea and pairs of compex ones and of course we get different conjugates Received by the editor August 30, 994 and, in revised form, March 27, 995 and Juy 20, Mathematics Subject Cassification. Primary Y40. Key words and phrases. Agebraic number theory, norm equations, reative norm equations, reative extensions. 399 c 997 American Mathematica Society License or copyright restrictions may appy to redistribution; see

2 400 C. FIEKER, A. JURK, AND M. POHST when we consider F as an extension of E rather than as an extension of Q. We wi discuss this in detai ater. The ring of integers of a fied K is denoted by o K in the seque. 3. Absoute norm equations In this section we give a short review on the Fincke Pohst agorithm to sove absoute norm equations. Given an arbitrary, but fixed k Z >0 and M EafreeZmodue of E, wewant to find a η M subject to (3.) or (3.2) N(η) =k N(η) = k, i.e., we want to determine if soutions exist and, if so, compute a of them. If M is of fu rank m, we fix a maxima system of independent units ɛ,..., ɛ r (r = m +m 2 ) of its ring of mutipiers. For practica computations it is advisabe to use LLL-reduction in the ogarithmic attice to produce units for which the absoute vaues of each of their conjugates are cose to. The next emma gives expicit bounds for a compete set of nonassociate soutions. Lemma 3.. Let η be a soution of (3.2). Then there exists a unit ɛ and a soution η = ηɛ subject to (3.3) where η(i) R m i k R i ( i m), R i := exp 2 m +m 2 j= og( ɛ (i) j ). Proof. See [, (6.3) Lemma] or [8, Theorem (4.2), Chapter 6]. If M is not of rank m, then it is not known how to compute reaistic bounds on the conjugates of the (finitey many) soutions of (3.2). Hence, in that case we need to stipuate bounds R i such that any soution η satisfies (3.3). This works we if we are ony interested in sma soutions, say with coefficients bounded by 0 6. Furthermore, we need the foowing, rather technica, emma from [, (6.23) Satz] or [8, Theorem (3.8), Chapter 5]: Lemma 3.2. For arbitrary γ, r R >0 define the functions h : R > R : t t t og t and g : R > R : t ( h(t))t h(t) + h(t)t h(t). Then there exists a unique zero λ = λ(γ,r) of g(t) ( + γ r )2/m. Using the previous two emmas, one can prove the foowing theorem, which wi aow us to sove absoute norm equations of the type (3.2) or (3.). Theorem 3.3. For arbitrary γ R >0 et λ = λ(γ,k) as in Lemma 3.2. Define constants L i := 2ogRi og λ, U i := 2ogRi og λ ( i m) with R i as in Lemma 3.. License or copyright restrictions may appy to redistribution; see

3 SOLVING RELATIVE NORM EQUATIONS IN ALGEBRAIC NUMBER FIELDS 40 If (3.2) is sovabe, then there exists r =(r,...,r m ) Z m and a soution η of (3.2) subject to (3.3), and m (3.4) λ rj η (j) 2 m(k + γ) 2/m. j= The coordinates of r satisfy m j= r j =0,L j r j U j ( j m) and r m+j = r m+m 2+j for a j m 2 with at most one exception, where we have r m+j + = r m+m 2+j. Forrank M = m, any soution of (3.2) is associate with one satisfying (3.4). Remark 3.4. Representing η in a basis of M, the eft hand side of (3.4) becomes a positive definite quadratic form over M, so that the soutions of the inequaity are attice points inside an eipsoid. The basic idea in the proof of the theorem above is an observation due to M. Pohst which guarantees (under certain conditions) the existence of some vector λ R m >0 with m j= λ j η (j) 2 = mk for every soution η of (3.2). The main task in the proof then is to obtain a finite set of candidates for the λ s. This was achieved by U. Fincke [] who transformed our probem into a discrete one by considering ony vectors of the form (λ r,...,λ rm ), r j Z. In the next step, we obtain bounds for the r j s from the bounds for the soutions. Finay, we reduce the set of the admissibe exponent vectors r =(r,...,r m ). To use this theorem for soving (3.2), one has to cacuate the set of a admissibe r Z m, for each such r to compute the set of a η M subject to (3.4) and, finay, to determine the soutions of (3.2) among them. The foowing statement concerns the agebraic compexity when using the Fincke-Pohst method to enumerate the points of the eipsoids. Theorem 3.5. Let k Z and the signature (m,m 2 ) be fixed. Then there exists a v > depending ony on the conjugates of a Z basis of M and a sufficienty arge U max m j=i U j, U j as in Theorem 3.3, such that the number of arithmetic operations (addition, mutipication, division, cacuation of square roots) used to sove (3.2) with the agorithm described above is bounded by (m 2 +)(3 4ogk mog λ + 4ogU og λ )m+m2 (2Um 2/3 vλ /2 (+ λ k )/m ) m/2 (2m 3 +24m 2 ). Proof. See [, (7.5) Satz]. 4. Reative norm equations Before we sketch the theory and the agorithm, we make some further definitions. Let M F be a free o E -modue. As above, we assume the rank of M to be n in order to produce bounds. For an arbitrary θ o E we want to decide if there are any η M satisfying (4.) N F/E (η) =θ or (4.2) N F/E (η) = θ and, if so, we wi compute a compete set of nonassociate soutions. Athough the agorithm described in the previous section coud, in principe, be used to accompish this (certainy one can sove N(η) = N E/Q (N F/E (η)) = License or copyright restrictions may appy to redistribution; see

4 402 C. FIEKER, A. JURK, AND M. POHST F ()... F (n)... F ((m )n+)... F (nm) E ()... Q E (m) Figure. The ordering of the conjugate fieds N E/Q (θ) and then test the soutions), it turns out to be far too expensive in terms of computation time. Hence, the approach for the absoute case needs to be changed appropriatey. To dea with the difficuties concerning the conjugates of F arising from the fact that we can consider F both as an extension of E and of Q, we fix the ordering of the conjugate fieds as described in Figure. To simpify notation, we define the foowing abbreviation: N (j) F/E (γ) :=N (γ ((j )n) )= n F ((j )n+) /E (j) i= γ((j )n+i). We require that the conjugate fieds E (),..., E (m) of E are ordered as described in 2. For the rea conjugate fieds E (i) ( i m ) we stipuate furthermore that the conjugate fieds of F, which are extensions of these, are ordered in a simiar manner. For i m we require that F ((i )n+j) R ( j s i )and F ((i )n+si+j) = F ((i )n+si+ti+j) R ( j t i ), s i +2t i =n,where(s i,t i ) denotes the reative signature of F ((i )n+) over E (i) [4, Lemmas ]. As in the absoute case, the first task is to obtain bounds for a set containing a non-associate soutions of (4.2) or (4.). In the previous section we used units of the ring of mutipiers of M to transform arbitrary soutions to those contained in a bounded region. In order to preserve (4.), one has to use units ɛ where N F/E (ɛ) =, whereas for (4.2), it suffices to require that N F/E (ɛ) is a torsion unit of E. Here we wi restrict ourseves to units of the form ɛ n /N F/E (ɛ). Ceary, we have N F/E (ɛ n /N F/E (ɛ)) = N F/E (ɛ n )/N F/E (ɛ) n =. Letɛ,..., ɛ r be a maxima system of independent units of the ring of mutipiers of M. We have the foowing emma, which corresponds to Lemma 3.: Lemma 4.. Let η be a soution of (4.2) or (4.). Then there exists a unit ɛ and a soution η = ηɛ of (4.2) and (4.) subject to η((j )n+i) (4.3) R R n (j )n+i ( i m, j n), (j )n+i θ (j) where R (j )n+i =exp ( n 2 r = og( ɛ ((j )n+i) ) + 2 ) r og( N (j) F/E (ɛ ) ). = License or copyright restrictions may appy to redistribution; see

5 SOLVING RELATIVE NORM EQUATIONS IN ALGEBRAIC NUMBER FIELDS 403 Proof. The proof is essentiay the same as in the absoute case. The bounds are worse than those in Lemma 3. because of the restricted set of units used for the transformation [4, Lemma 4.]. Assuming that we have bounds for the conjugates of soutions from a different source, we can again generaize the method to modues M of rank ess than n, as in the ast section. We note that the bounds here are not a generaization of those obtained in Lemma 3.. Having estabished this emma, and using Lemma 3.2 of the preceding section, we obtain the foowing theorem: Theorem 4.2. For arbitrary γ j R >0 et λ j = λ(γ j, θ (j) )( j m)as in Lemma 3.2. Define constants L (j) i := 2ogR (j )n+i og λ j, U (j) i := 2ogR (j )n+i og λ j ( i n) with R (j )n+i as in Lemma 4.. Then for every soution ν of (4.) there exists a unit ɛ with N F/E (ɛ) =,r (j) =(r (j),...,r(j) n ) Zn ( j m) and a soution η = ɛν of (4.) subject to (4.3) such that () n i i= λr(j) (2) m i= r(j) i (3) L (j) i j η ((j )n+i) 2 n( θ (j) + γ j ) 2/n, j m; =0, j m; U (j) i, i n, j m; r (j) i (4) For j m we have in addition r (j) s j+i = r(j) s j+t j+i for a i t j with at most one exception, where we have r (j) s j+i +=r(j) s. j+t j+i Proof. Let ɛ and η = ν be as in Lemma 4.. For j m and i n define y (j) i := θ (j) 2/n η ((j )n+i) 2. Ceary, m i= y(j) i =. Sincey (j) i and λ j are positive, we can find r (j) i Z, ẽ (j) i [0, ) with (4.4) y (j) i = λ r(j) i j and 0 = n = r(j) = n = ẽ(j). By performing some engthy computations, we can change the ẽ s and the r s to fufi conditions (2) and (4) and then verify (3) using (4.4). As in the absoute case, the vaidity of () is a consequence of Lemma 3.2 [4, Satz 4.4]. A straightforward computation gives N (j) F/E (η) θ(j) +γ j for a η satisfying () of Theorem 4.2. Anaogous to the description given in the paragraph after Remark 3.4, we get an agorithm for soving (4.2). We note that the inequaities () describe eipsoids defined via the positive definite quadratic forms [4, Lemma 4.6] x =(x,...,x n ) n i= +ẽ (j) i λ r(j) i j x j 2 ( j m). Their attice points can be cacuated with a modified Fincke-Pohst enumeration agorithm. 5. A modified Fincke-Pohst agorithm A main part in soving norm equations is the enumeration of a points in suitabe eipsoids. In this section we present a modification of the Fincke-Pohst method License or copyright restrictions may appy to redistribution; see

6 404 C. FIEKER, A. JURK, AND M. POHST adopted to reative norm equations. We consider the foowing situation: Let A (i) C n n be positive definite, C (i) > 0 subject to A (i) = A (i+m2), C (i) = C (i+m2) (m <i m +m 2 ). We present an agorithm to cacuate a x M subject to x tr (i) A(i) x (i) C (i), where x (i) =(x ((i )n+),...,x ((i )n+n) ) tr. Let ν,...,ν n be an o E -basis of M. For x M we have x= n = z ν with z i o E and x ((i )n+j) = n = z(i) ν ((i )n+j). Defining V i := (ν ((i )n+j),weset ) j n n Q i (x):=x tr (i) A(i) x (i) =(z,...,z n )V tr i A (i) V i (z,...,z n ) tr. Let B i := V tr i A(i) V i. Using the agorithm for quadratic suppement (see [3, (2.3)]), we obtain z tr B i z = n = q(i), z(i) + n s=+ q(i),s z(i) s 2. (Note that we have q (i), > 0 since B i is positive definite.) This yieds the foowing straightforward agorithm: Agorithm 5.. (Reative enumeration) Input: q (i),s, C(i) as above Output: A x M subject to Q i (x) C (i) Init: Step : := m, T (i) := C (i),u (i) S := {z o E z (i) U (i) := 0 ( i m). 2 T(i) i m}. Step 2: If S goto Step 4. Step 3: Set := +,if>mthen terminate ese goto Step 2. Step 4: Choose z in S arbitrary and deete z from S. Step 5: If =printx:= n s= z sν s goto Step 2. Step 6: :=, T (i) := T (i) + q(i) +,+ (z(i) + + U (i) (i) + ), U := n s=+ q(i),s z(i) s, ( i m) goto Step. It remains to give an agorithm to compute the set S in Step. We wi do this using a modified Fincke-Pohst agorithm. Instead of computing S, we compute a set S := {z o E m i= z(i) U (i) 2 C } with C := m T (i) i=.letµ q (i),...,µ m, be a Z-basis of o E and define µ ()... µ () m U () W :=... Cm (m+). µ (m)... µ (m) m q (i), U (m) Now we have m i= x(i) U (i) 2 =(z,...,z m,)w tr W (z,...,z m,) tr for every x = m s= z sµ s o E, z s Z. Ceary, W tr W is positive semidefinite and rea, so the S can be cacuated with the cassica Fincke-Pohst agorithm if we fix the ast coordinate to be. License or copyright restrictions may appy to redistribution; see

7 SOLVING RELATIVE NORM EQUATIONS IN ALGEBRAIC NUMBER FIELDS The agorithm We are now abe to present a compete agorithm for soving (4.2). Agorithm 6.. (Computing a compete set of nonassociate soutions of (4.2)) Input: θ o E, γ R m >0, M and a compete set ɛ,...,ɛ r of independent units of the ring of mutipiers of M. Output: A compete set of nonassociate soutions of (4.2). Init: Compute λ j = λ(γ j, θ (j) ) as in Lemma 3.2, R (j )m+i as in Lemma 4. and L (j) i and U (j) i as in Theorem 4.2. Let S :=. Step : Compute I := {(r (j) i ) i n Z n m (2) (4) of Theorem 4.2 hod }. j m Step 2: Whie (I ) do Init enumeration: Choose r I arbitrary, and deete r from I. Enumerate: Determine a η M satisfying () of Theorem 4.2 (e.g., using Agorithm 5.). Check: For each soution of the previous step, check (4.2). If we have a soution, check whether it is associated with one aready in S. If we have a new soution, store it in S. Step 3: If S = return No soution ese return S. In order to sove (4.) instead of (4.2), one simpy has to change the condition in Check. To estimate the agebraic compexity of this agorithm, we introduce the T 2 -norm: T 2 : E R:η m i= η(i) 2 and the so caed reative T 2 -norm T F/E 2,j : F R:η n i= η((j )n+i) 2 ( j m). Let ν,..., ν n be an o E -basis of M and ν,..., νn the corresponding dua basis subject to Tr F/E (ν i νj )=δ i,j. Define d j := max T F/E 2,j (νi )( j m), L j := max{λ r(j) i j i n m C := d j L j (n θ (j) + γ j ) 2/n. j= r I, i n}, Furthermore, et µ,...,µ m be a Z-basis of o E and µ,..., µ m its dua basis subject to Tr E/Q (µ i µ j )=δ i,j. Letd:= max m T 2 (µ ). We begin with the compexity of Agorithm 5.. Lemma 6.2. () The ( agorithm for computing one set S i in Step of Agorithm 5. cacuates O (2 ( ( 4dC +m ) )) dc +) 4dC + points. For each point it needs O(m 2 ) operations, ( and additiona ( O(m 3 ) operations for precomputing. (2) Agorithm 5. needs O (mn 2 + m 3 ) (2 ( ( 4dC +m ) )) n ) dc +) 4dC + operations. Proof. In the same way as in [3, Proof of 3.5] (cf. [4, Lemma 3.7] or [4, Lemma 4.3]) we normaize the task to q (i),. Now C is an upper bound for a S i. () Since we fix the ast coordinate, we effectivey enumerate an m dimensiona eipsoid, so the statements foow at once from [3]. License or copyright restrictions may appy to redistribution; see

8 406 C. FIEKER, A. JURK, AND M. POHST (( (2) Ceary, we consider at most O (2 dc +) and for each point we need O(mn 2 + m 3 )operations. ( ( 4dC +m 4dC ) )) n ) + points, We note that we need O(mn 3 ) operations for cacuating the matrices B i and the q (i),s. Theorem 6.3. Define D j := 2 max i n og R (j )n+i, R (j )n+i as in Lemma 4., j m, ( Q := mn 3 +(mn 2 + m 3 ) (2 ( ) n 4dC + m dc +) +). 4dC The set I := {(r (j) i ) i n j m Z n m (2) (4) of Theorem 4.2 are vaid } contains at most m m 2 +m 2 ((t j +)( D j +3) sj+tj 2 ) ( D j +3) n og λ j og λ j j= j=m + eements. For each (r (j) i ) i n I (i.e., for each quadratic form considered) the j m agorithm needs O(Q) arithmetic operations to enumerate a η with () of Theorem 4.2. Hence Agorithm 6. requires arithmetic operations. O(( I)Q) Proof. See [4, Lemmas 4.2 and 4.3]. We note that the estimate for I is a straightforward computation using ony (2) (4) of Theorem 4.2 and Lemma 4.. Let E := Q(α), α a zero of 7. Exampe f(x) :=x 3 +2x 2 x+2. We consider the extension F := E(β) of absoute degree 9, where β is a zero of g(x) :=x 3 +x 2 +(+2α+α 2 )x+( 2+α 2 ). The signature of E is m = m 2 =. SinceFhas ony one rea embedding, the unit rank of F is 4. A system of fundamenta units of o F is ɛ := 2 (α + α2 ), ɛ 2 := 2 (2 + 2α +(2 α α2 )β), License or copyright restrictions may appy to redistribution; see

9 SOLVING RELATIVE NORM EQUATIONS IN ALGEBRAIC NUMBER FIELDS 407 ɛ 3 := ( 3+α 2 +(2 2α α 2 )β+(+α)β), ɛ 4 := 2 ( α +5α2 +(6+29α+9α 2 )β+(4 9α 9α 2 )β 2 ). We consider the foowing probems: () Sove N F/Q (x) = k for x o F, k Z >0, using the absoute method. From Lemma 3. and Theorem 3.3 we obtain bounds (after an LLL-reduction of the unit attice): i L i = U i This requires us to consider eipsoids if a soutions of the norm equation are to be determined. (2) Sove N F/E (x) = θ for x o F and θ o E by (a) the absoute method: Lemma 4. and Theorem 3.3 yied bounds: i L i U i A tota of eipsoids need to be enumerated. (b) the reative method: Over E () we have the reative signature t = s =. By Lemma 4. and Theorem 4.2 we obtain: j 2 3 i L (j) i = U (j) i that is a tota of ony 4289 eipsoids. We note that we actuay computed a these eipsoids, so that the numbers L i, U i given above are exact. 8. Tabes A computations were carried out on an HP9000/735s with 96MB memory using software deveoped under KANT V4 [5]. The operating system on the machine was HP-UX 9.0. F. Grunewad in Düssedorf asked us the foowing question: Let E = Q(α), α {, 2, 3},β= a+bα for 3 a, b 3 such that F = E(β) isof absoute degree 4. How many nonassociate soutions do exist for (4.2) with sma θ, where nonassociate means moduo units of reative norm? In Tabes and 2 we can ony present a part of our resuts because of imited space. Another probem of W. Pesken in Aachen and H. Brückner in Hamburg was to sove severa norm equations in cycotomic fieds (see Tabe 3 for detais). We present soutions of (4.), meaning that no soutions exist, and ζ k denoting a primitive kth root of unity. License or copyright restrictions may appy to redistribution; see

10 408 C. FIEKER, A. JURK, AND M. POHST Tabe. Exampes E F θ soution re. norm Q( 2) E( ) 2 ( ) (2 2( + )) 2 2 (2 + 2( + )) 2 ± 2 3 ± (2 ± ( ) 2 2 ) ± 2 2 Q( 3) E( ) (± 3) 2 ( 3) 2 ( 2+ (± 3)) 2 ( 3) 2 ( + 3 ( + 3)) 2 ( (± 3)) (2 3+ (± 3)) 2 ( 7 3) 2 ( ) 2 (5 3 3) 2 (4 + ( +3 3)) 2 ( 5 3 3) 2 ( ( 2+2 3)) 2 (7 + 3) ( ) 2 (7 3) ( + 3) 2 3 License or copyright restrictions may appy to redistribution; see

11 SOLVING RELATIVE NORM EQUATIONS IN ALGEBRAIC NUMBER FIELDS 409 Tabe 2. Exampes E F β soution re. norm Q( ) E( 2) + 2 2(± ) ± 2 (2 + 2( + )) ± 2 (±2+ 2(± )) ± 2 ± (± 3 ) 4 ± 3 2( 3 ± ) 4 ± 3 ±2 3 4 ± p E( ) p p + p + + p p + E( p + ) p + + p + + p + p + p p + License or copyright restrictions may appy to redistribution; see

12 40 C. FIEKER, A. JURK, AND M. POHST Tabe 3. Exampes E F β soution Q(ζ 5 ) R Q(ζ 5 ) 2 ζ 4 ζ 5 ζ 5 Q(ζ 7 ) R Q(ζ 7 ) 2 ζ 7 ζ 7 ζ 7 Q(ζ 8 ) R Q(ζ 8 ) 2 ζ 8 ζ 8 ζ 8 Q(ζ 9 ) R Q(ζ 9 ) 2 ζ 9 ζ 9 ζ 9 Q(ζ 0 ) R Q(ζ 0 ) 2 ζ 0 ζ 0 ζ 0 Q(ζ ) R Q(ζ ) 2 ζ ζ ζ Q(ζ 2 ) R Q(ζ 2 ) 2 ζ 2 ζ 2 ζ 2 Q(ζ 6 ) R Q(ζ 6 ) Q(ζ 32 ) R Q(ζ 32 ) Q(ζ 64 ) R Q(ζ 64 ) References. U. Fincke, Ein Eipsoidverfahren zur Lösung von Normgeichungen in agebraischen Zahkörpern, Thesis, Düssedorf U. Fincke, M. Pohst, A Procedure for Determining Agebraic Integers of Given Norm, Proceedings EUROCAL 83, Springer Lecture Notes in Computer Science No. 62, Springer, 983. MR 86k: U. Fincke, M. Pohst, Improved methods for Cacuating Vectors of Short Length in a Lattice, Incuding a Compexity Anaysis, Math. Comp. 44, (985). MR 86e: A. Jurk, Über die Berechnung von Lösungen reativer Normgeichungen in agebraischen Zahkörpern, Thesis, Düssedorf The Kant-Group, KANT V4, to appear in J. Symb. Comput. 6. K. Maher, Inequaities for Idea Bases in Agebraic Number Fieds, J. Austra. Math. Soc. 4, (964). MR 3: W. Narkiewicz, Eementary and Anaytic Theory of Agebraic Numbers, 2nd ed., Springer, 990. MR 9h:07 8. M. Pohst, H. Zassenhaus, Agorithmic Agebraic Number Theory, Encycopaedia of Mathematics and its Appications, Cambridge University Press, 989. MR 92b:074 Fachbereich 3 Mathematik, Sekretariat MA 8, Technische Universität Berin, Straße des 7. Juni 36, D-0623 Berin, Germany E-mai address: fieker@math.tu-berin.de Desdorfer Weg 5, 508 Bedburg, Germany Fachbereich 3 Mathematik, Sekretariat MA 8, Technische Universität Berin, Straße des 7. Juni 36, D-0623 Berin, Germany E-mai address: pohst@math.tu-berin.de License or copyright restrictions may appy to redistribution; see