c
de Gruyter 2009
J. Math. Crypt. 3(2009), 237–248 DOI 10.1515 / JMC.2009.014
On solving norm equations in global function fields
István Gaál and Michael E. Pohst
Communicated by Jaime Gutierrez
Abstract. The potential of solving norm equations is crucial for a variety of applications of algebraic
number theory, especially in cryptography. In this article we develop general effective methods for
that task in global function fields for the first time.
Keywords. Norm equations, unit equations, global function fields.
AMS classification. 11Y50, 11T71, 11D57.
1 Introduction
Let E/F be a (finite) extension of fields of degree d. Let ω1, . . . , ωdbe a fixed F-basis
of E. Then each element x∈Ehas a presentation
x(ω1, . . . , ωd)=(ω1, . . . , ωd)Mx
with a matrix Mx∈Fd×d. The determinant det(Mx)is called the norm N(x)of the
element x∈Ewith respect to F. Since N(x)is – up to sign – the constant term in the
characteristic polynomial of xit is independent of the choice of the basis.
The calculation of elements of prescribed norm is an important task in algebraic
number theory.
From now on, we consider the situation in which the base field Fis either the field of
rationals Qor a rational function field k(t)over a finite field k=Fqof characteristic p,
say with q=p`. Let oFbe the ring of integers of F, i.e. Zor k[t], respectively.
The extension Eof Fis assumed to be separably generated by an element ywith
minimal polynomial my(T)∈oF[T]. Then there are ddifferent embeddings of Einto
the algebraic closure ¯
F, say σ1, . . . , σd. It can be easily shown that
N(x) =
d
Y
j=1
σj(x)∀x∈E .
In the context of global fields, norm equations are in general discussed as follows.
We choose 2 ≤m≤d F-linearly independent elements γ1, . . . , γmof E, usually in
oE, the integral closure of oFin E.
Then M=⊕m
i=1oFγiis a free oF-module in E. We are going to look for solutions
of
N(x) = c(x∈M)
Research of first author supported by K67580 and T048791 from the Hungarian National Foundation for
Scientific Research
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238 István Gaál and Michael E. Pohst
for given cof oF. Writing x=x1γ1+· · · +xmγmwith xi∈oF, the norm N(x)
becomes a form in the xiwith coefficients in F.
For two special values of mthe computation of solutions is quite well understood.
The first one is m=dand the module Mis an order of E. This means that the unit
group UMof Moperates on M. If there is a solution of N(x) = cthen also xε is one
for any ε∈UMof norm 1.
Multiplying xwith a suitable ε, we can impose conditions on the conjugates σj(x).
All elements of Msatisfying those conditions can then be calculated by reduction
theory. These ideas are well known for algebraic number fields, see [3], [11], for
example. For global function fields the reduction procedure is more subtle because
there does not exist an analogue of the LLL-algorithm. We discuss that reduction
procedure in Section 3.
We note that both methods require the computation of a set of generators of UE, the
unit group of oE. The latter is possible with KANT or Magma, for example. (In the
number field case also by Pari, of course.)
The second well understood case is m=2 and d > 2. In this situation N(x)
becomes a binary form and N(x) = ca so-called Thue-equation. It can be reduced
to a unit equation in two unknowns as follows. For simplicity’s sake we assume γ1=
1, γ2=α, hence x=x1+x2α. Then we have Siegel’s identity:
(σi(α)−σj(α))σk(x) + (σj(α)−σk(α))σi(x) + (σk(α)−σi(α))σj(x) = 0
for any three pairwise different indices 1 ≤i, j, k ≤d. Dividing by the last summand,
we obtain
(σi(α)−σj(α))σk(x)
(σi(α)−σk(α))σj(x)+(σj(α)−σk(α))σi(x)
(σi(α)−σk(α))σj(x)=1.
The last equation becomes an S-unit equation u1+u2=1 by writing x=µε with
ε∈UEand µ∈oEfrom a finite set of non-associate solutions of N(µ) = c. (Note
that S-units are defined at the beginning of the next section.)
In thenumber fieldcase oneuses Baker typeresults onlinear formsin logarithms, re-
duction theory and refined enumeration strategies for computing solutions. The fastest
algorithm known is that of Bilu and Hanrot [1]. The function field case is discussed in
detail in [5]. We just sketch the method below.
According to our present knowledge the resolution of unit equations in more than
two variables is needed for solving norm equations in 2 < m < d variables. It is
not known how to do this in number fields. Therefore it is somewhat surprising that
a resolution is still possible in function fields. In [6] we considered the case m=3
which we now generalize to arbitrary m. Also, the case m=dwill shortly be treated
since we came up with some improvements over the known methods. We conclude
with a small illustrative example.
In recent years, the construction of elliptic curves suitable for pairings has attracted
increasing interest. The construction of Weil numbers in CM-fields, suitable for pair-
ings, leads to systems of diophantine equations. In a current project we will apply the
methods of this paper to obtain solutions in integers as well as polynomials.
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On solving norm equations in global function fields 239
2 Global fields: unit equations
Unit equations of type
u1+· · · +un=1,
where the uiare elements in a unit group of a function field, play an essential role in the
theory and in applications of diophantine equations. We introduce several important
notations.
Let k=Fqbe a finite field with q=p`elements. Let Ebe a finite extension of the
rational function field F=k(t)of degree dand genus g. The integral closure of k[t]in
Eis denoted by oE. We assume that Eis separably generated over k(t)by an element
y∈oEand that kis the full constant field of E.
Any element f∈Ehas a unique presentation
f=
d
X
i=1
hiyi−1(hi∈k(t)) .
Conjugates of elements (fields) are denoted by upper case indices, i.e. we write x(j)for
σj(x)and E(j)for σj(E). Let
A:=(y(j))i−11≤i,j≤d∈Ed×d
have determinant D. Since Eis separably generated we have D6=0.
From the system of linear equations
(f(1), . . . , f(d))=(h1, . . . , hd)A
we conclude that the hiare rational functions in the f(j),(y(j))i−1.
The set of all (exponential) valuations of Eis denoted by V, the subset of infinite
valuations by V∞. We write degvfor the degree of the divisor belonging to the valu-
ation v∈V. For a non-zero element f∈Ethe value of fat vis denoted by v(f).
For integral elements this is the highest power of the divisor belonging to vthat divides
the divisor (f), and this concept is extended to rational elements in the usual way. The
normalized valuations vN(f) = v(f)degvsatisfy the product formula:
X
v∈V
vN(f) = 0∀f∈E\ {0}.
The height of a non-zero element fof Eis defined via
H(f):=X
v∈V
max{0, vN(f)}.
Because of the product formula this is tantamount to
H(f) = −X
v∈V
min{0, vN(f)}(f∈E).
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240 István Gaál and Michael E. Pohst
Let V0be a finite subset of V. We then consider the V0-units γ∈Ewith v(γ) = 0
for all v6∈ V0.
As we saw in the introduction, the resolution of various Diophantine equations, for
example, Thue equations, can often be reduced to that of equations of the form
γ1+γ2+γ3=0,
where the γiare V0-units for a suitable set V0. We excerpt the following crucial lemma
from [4].
Lemma 2.1. Let V0be a finite subset of Vand let γi(1≤i≤3)be V0-units the sum
of which equals 0. Then either γ1
γ3is in Epor its height is bounded:
Hγ1
γ3≤2g−2+X
v∈V0
degv .
Setting Φ=−γ1/γ3,Ψ=−γ2/γ3, we obtain a unit equation in two variables
Φ+Ψ=1,
with V0-units Φ,Ψ. Because of the characteristic pthe number of solutions of such a
unit equation is in general infinite.
For example, if V0is just the set of infinite valuations and η, 1−ηare both units of
oE\k, then also ηκ,(1−η)κis a solution for every exponent κ=pτ. Hence, there
exist solutions of arbitrary large heights in this situation.
The subsequent corollary shows that for any finite subset V0of Vthe group of V0-
units of Econtains only a finite number, say σ, of V0-units ηwhich are not pτth powers
and for which also 1 −ηis a V0-unit. We denote the set of these units by {η1, . . . , ησ}.
Corollary 2.2. Let V0be a finite subset of V. We assume that a V0-unit Φin Eis
a solution of a unit equation in two variables. If Φis not a pτth power of an element
ηi(1≤i≤σ, τ ∈Z>0), then Φbelongs to a finite subset of Ewhich can be effectively
calculated.
For the proof we refer to [7, Page 98, Lemma 11]. The proof starts by assuming that
Φis not a pth power in Kand therefore also provides the means to calculate the ηi. For
an example of a Thue equation with infinitely many solutions see [5].
Next we consider unit equations in more than two variables. Again, V0denotes
a finite subset of Vcontaining the infinite valuations. Let γi(1≤i≤n)be V0-units.
The equation
γ1+. . . +γn=0 (2.1)
is equivalent to the unit equation
−γ1
γn+· · · +−γn−1
γn=1 (2.2)
in n−1 variables. We note that we only need to postulate that all fractions in the last
equation must be V0-units.
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On solving norm equations in global function fields 241
Those unit equations are of importance since several well-known Diophantine equa-
tions (e.g. norm form equations) can be reduced to unit equations in more than two
variables. From the results in [6] we easily deduce the following theorem.
Theorem 2.3. Let V0be a finite subset of Vand let γi(1≤i≤n)be V0-units satis-
fying (2.1). Assume that no proper subsum of the sum in (2.1) vanishes. Then we can
explicitly construct a finite subset Nof V, such that
γi
γn
=xin ·Φi,(2.3)
where xin is a solution of a unit equation
x1n+x3n+· · · +xn−1,n =1
with V0∪N-units xin (i=1,3, . . . , n −1), and a V0∪N-unit Φisatisfying
H(Φi)≤2g−2+X
v∈V0
degv . (2.4)
Hence, the solution of a unit equation in n−1V0-units is reduced to determining
the solutions of unit equations in n−2V0∪N-units. In [6] only the case n=4 was
considered. The generalization to arbitrary nis done here for the first time.
If we replace the γi/γnin the original unit equation (2.2) via (2.3), then we get
n−1
X
i=1
i6=2
xinΦi=−1.(2.5)
We need to consider several cases. The main ingredient is to deduce solutions of
unit equations from those of unit equations in fewer variables. We remark that this
discussion does not include the case in which all xin,Φiare in k(compare also the
premises in Theorem 2.3).
I. If any of the elements xin (i=1,3, . . . , n −1), say x1n, is not a pth power, then
they are obtained from a finite set of solutions of an S-unit equation in n−2 variables
(compare Theorem 2.3). Also, the Φ1, . . . , Φn−1can attain only finitely many values
by that theorem. This reduction needs to be carried out until we get unit equations in
at most 3 variables, a case which is already treated in [6].
II. If all xin (i=1,3, . . . , n −1)are pth powers, then using local derivation at an
arbitrary valuation we get from (2.5) equations
n−1
X
i=1
i6=2
xinΦ[j]
i=0(j∈N),(2.6)
where Φ[j]
idenotes the jth derivative of Φi. We note that the derivatives of the xin
vanish in this case. We need to discuss subcases (A), (B).
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242 István Gaál and Michael E. Pohst
(A) If all Φi(i=1,3, . . . , n −1)are pth powers (especially, if they all are in k),
then equation (2.6) is meaningless but both sides of (2.5) are pth powers and we can
take pth roots to make the valuations of the xin,Φismaller. This can be applied re-
peatedly until we end up in case I.
(B) Without loss of generality, we assume Φ16∈ kwith Φ[1]
16=0. Because of our
assumption, in this case we have at least one index µ > 1 with Φ[1]
µ6=0. We compute
equations (2.6) for j=1,2, . . . and obtain a linear system of equations for the xin. If
there are n−2 independent equations for the xin then that system has only the trivial
solution and there does not exist a solution of (2.5) for this case. Otherwise, those
equations can be used for eliminating variables. We end up in case II but with fewer
variables.
3 Application to norm form equations
As before, let Ebe a finite extension field of F=k(t)of degree d≥3 and denote
by oEthe integral closure of k[t]in E. Then oEhas an oF-basis (integral basis), say
α1, . . . , αd.
3.1 Norms from oE
This subsection improves the results of [10]. In that thesis the reduction theory of
W. Schmidt in characteristic 0 was generalized to arbitrary characteristics. However,
the case when the infinite prime is wildly ramified remained open. In [8] we also
showed how to treat that case. Here, we shortly discuss the whole reduction procedure.
Let α∈oEbe a solution of NE/F (α) = cfor c∈oF. Then also αε is a solution for
any unit εof the unit group UFwith NE/F (ε) = 1. We are therefore only interested
in non-associate solutions of the original norm equation, i.e. solutions which do not
differ by a unit. Scaling such a solution with a power product of the generators of UE
(fundamental units) we obtain a solution in an appropriate finite dimensional F-vector
space. For this we need a system of fundamental units ε1, . . . , εs−1and a reduced basis
for oE, where sdenotes the number of infinite primes. A system of fundamental units
can be computed with Kash or Magma. The full reduction procedure was originally
developed in [8]. We sketch the ideas.
We make use of the fact that the non-zero elements vof Eadmit series expansions
(Puiseux series, respectively, Hamburger–Noether series) of the form
Σ(v):=
∞
X
i=m
φit−νi(3.1)
with rational exponents νm< νm+1< . . . and coefficients φi∈F×
q. (We note that
this is a simplification, in general the φimay belong to a small finite extension of Fq
of degree e, where eis the least common multiple of the ramification indices of the
infinite primes in E.) We need to assume that Eover Fis separately generated (which
can always be achieved by the choice of F). We then have E=F(y)for a suitable
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On solving norm equations in global function fields 243
element ywhose minimal polynomial in oF[t]has ddifferent zeros. Each zero admits
a series expansion so that we obtain a total of dseries expansions corresponding to the
conjugates of Eover F. The isomorphic embeddings of Einto the algebraic closure
¯
Fare denoted by σ1(y), . . . , σd(y), as usual. In each case we obtain an exponential
valuation on Evia
ord : F →Q: v 7→ νmfor v 6=0,
∞for v =0.(3.2)
We denote the exponential valuations belonging to σ1, . . . , σdby ord1, . . . , ordd, re-
spectively.
Now let B={v1, . . . , vd}be a basis of a non-zero ideal Aof oE. We denote by vj=
(vij)1≤i≤dthe vector whose components vij are the dseries expansions σi(vj) (1≤
i≤d). We set ord(vj):=minordi(vij): 1 ≤i≤d=:νj.
For each basis element vjwe therefore obtain a vector Φj= (φij)1≤i≤dof coefficients
of the leading term of the series expansions, i.e. φij is the coefficient of t−νjin the
expansion σi(vj).
We then set ord(A):=orddet(vij)1≤i,j≤d
(in the sense of (3.2)) and
ψ(B):=ord(A)−
d
X
j=1
ord(vj).
The basis Bis called reduced if the non-negative value ψ(B)vanishes. We have
Lemma 3.1. An ideal basis B={v1, . . . , vd}of an ideal Ais reduced if and only if
for all (f1, . . . , fd)∈Fq[t]n,
ordd
X
i=1
fivi=min
1≤i≤dord(fivi).
If a basis {v1, . . . , vd}of an ideal Ais reduced and ordered subject to −ord(v1)≤
−ord(v2)≤. . . ≤ −ord(vd), then the values −ord(vi)are the successive minima of the
ideal A.
In [8] we developed an algorithm for computing a reduced basis which runs in poly-
nomial time in the input data.
From nowon, we stipulatethat weknow areduced basisω1, . . . , ωdof oE. Itis of im-
portance for calculating fundamental units of E[10] and of elements of bounded max-
imum norm (see below) which we need for calculating a maximal set of non-associate
solutions of NE/F (α) = cfor c∈oF.
In the following, the places of Fare denoted by lower case boldface letters, those
of Eby upper case boldface letters. The infinite place of Fwhich corresponds to
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244 István Gaál and Michael E. Pohst
the degree valuation is written as p∞. For P|pthe integers eP|p,fP|pand nP|p=
eP|pfP|pdenote the ramification index, the residue class degree and the local degree,
respectively. N(P)is the number of elements in the residue class field of P. The
exponential valuation belonging to Pis denoted by νP. For every element f∈Ewe
then set
|f|P:=N(P)−νP(f)/nP|p.
This normalization has the effect that |·|Pis a prolongation of |·|pand that the product
formula is still valid.
Definition 3.2. The maximum norm of an element f∈Eis defined by
kfk∞:=max
P|p∞
|f|P.
We note that according to [10] a reduced basis ω1, . . . , ωdof oEsatisfies
kfk∞=max|λi|∞kωik∞: 1 ≤i≤n
for any f=Pd
i=1λiωi∈oE. (This is the analogue of the previous lemma.)
We want to compute a full set of non-associate solutions of NE/F (α) = cfor c∈oF.
We improve the known methods for number fields and adapt them to the function field
case. For number fields the conjugates α(j)of a solution αof NE/F (α) = |c|satisfy
log
α(j)
c1/n =
r
X
i=1
xilog|ε(j)
i|(xi∈R,1≤j≤d)
for a full set of fundamental units ε1, . . . , εrof oE. (We note that r=s−1 and that a
full set of independent units suffices.) In the function field case we let Pbe one of the
infinite primes of Eand obtain for a solution αof NE/F (α) = cµ for arbitrary µ∈k
analogously
νP(α) =
r
X
i=1
xiνP(εi) + 1
dνp(c).
We note that the coefficients xiare independent from the choice of P|p. Substituting α
by an associate element just changes the coefficients xiby rational integers. Such a
substitution is supposed to yield small bounds Bfor logα(j)
c1/n , respectively for
|α|P=N(P)−νP(α)
nP|p
and hence for kαk∞. This amounts to calculate the maximum distance of an element
in the fundamental parallelotope of the lattice spanned by the vectors
log|ε(j)
i|1≤j≤d(i=1, . . . , r),
respectively
νP(εi)P|p(i=1, . . . , r),
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On solving norm equations in global function fields 245
to any lattice point. That task does not cause problems in the dimensions rfor which
independent (fundamental) units can be presently calculated. Knowing Bthe calcula-
tion of the corresponding solutions αof the norm equation is straightforward by the
methods in [9], respectively, in the function field case Byields bounds for the height
of α.
We note that the methods of this subsection also hold for arbitrary orders of E, not
just the maximal order oE.
3.2 Norms from free modules of degree less than d
We let 2 ≤m < d and αi(i=1, . . . , m)be F-linearly independent elements of oE
with E=F(α1, . . . , αm). For 0 6=µ∈oFwe consider the norm form equation in m
variables
NE/F m
X
i=1
xiαi=µ(xi∈oF,1≤i≤m).(3.3)
We recall that α(j)
i(j=1, . . . , d)denote the conjugates of αiover F. For any distinct
indices 1 ≤i1< i2< . . . < im≤d, the linear forms
δij(X1, . . . , Xm) =
m
X
i=1
Xiα(j)
i
are linearly independent over E. It is easy to calculate γij∈oE(1≤j≤m)such that
any solution x1, . . . , xnof equation (3.3) satisfies
m
X
j=1
γijδij(x1, . . . , xm) = 0.
Dividing by the last summand on the left-hand side, we obtain an S-unit equation in
m−1 variables:
−
m−1
X
j=1
γijδij(x1, . . . , xm)
γimδim(x1, . . . , xm)=1.(3.4)
Let V0be the set of valuations of Econtaining the infinite valuations and the valuations
occurring in µ. Then all δij(x1, . . . , xm)are V0-units. Let V1be an extension of the set
V0containing also the valuations occurring in any of the γij∈E(j=1, . . . , m). Then
all fractions in equation (3.4) are V1-units and we can apply the results of the previous
section. Usually pth powers can be excluded if there exist valuations (not contained
in V0) which only occur in γij/γimwith values not divisible by p, but cannot occur in
δij(x1, . . . , xm)/δim(x1, . . . , xn)or by considering Galois automorphisms (see [6]).
In this way we can calculate elements νijsuch that
δij(x1, . . . , xm) = νijδim(x1, . . . , xm)
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246 István Gaál and Michael E. Pohst
for j=1, . . . , m. Substituting them into equation (3.3), we get
m
Y
j=1
νijδim(x1, . . . , xm)m=µ ,
from which we calculate δim(x1, . . . , xm)and subsequently all δij(x1, . . . , xm). By
solving systems of linear equations, we can then determine all possible solutions
(x1, . . . , xm)of equation (3.3).
Remark. We note that increasing the number of variables in the unit equation makes
our procedure less efficient since the number of valuations usually increases drastically.
The amount of calculations to be performed grows roughly exponential with the growth
of the number of variables in the unit equation (see the construction in Theorem 2.3).
4 Example
Let k=F5and let α=α1be a root of
p(z) = z5−z−t=0.
Let E=k(t)(α)and denote by oEthe integral closure of k[t]in E. The field Ehas
genus g=0. This field is a Galois (Artin–Schreier) extension, the cyclic Galois group
is generated by σ:
αi+1=σ(αi) = αi+1(i=1,2,3,4).
Let rbe a non-zero constant (in k) and consider the solutions of the equation
NE/k(t)(x1+αx2+α2x3+α3x4+α4x5) = rin x1, x2, x3, x4, x5∈k[t].(4.1)
Set `i(x) = x1+αix2+α2
ix3+α3
ix4+α4
ix5, then we have the identity
`1(x) + `2(x) + `3(x) + `4(x) + `5(x) = 0.
The function field Ehas one infinite valuation v∞of degree 1. Let V0={v∞}. Then
for any solution x= (x1, x2, x3, x4, x5)∈(k[t])5of equation (4.1), the terms in
−`1(x)
`5(x)−`2(x)
`5(x)−`3(x)
`5(x)−`4(x)
`5(x)=1
are V0-units. (We note that in case of zero subsums we still get the same solutions.)
By [6] we have
`i(x)
`5(x)=xi·Φi,
where xi,Φiare V0∪N1-units,
x1+x2+x3=1,(4.2)
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On solving norm equations in global function fields 247
and
H(Φi)≤2g−2+X
v∈V0∪N1
degv .
Constructing the set of valuations N1, we find that Pv∈N1degvcan be estimated from
above (see [6]) by
2g−2+X
v∈V0
degv=−1,
hence N1is empty. Therefore xi,Φiare V0-units, and by H(Φi) = 0 the Φiare
constants.
We describe now the solutions of (4.2). By [6] we have
xi=yi·Ψi,
where yi,Ψiare V0∪N1∪N2-units, y1, y2are solutions of the unit equation y1+y2=1
and
H(Ψi)≤2g−2+X
v∈V0∪N1∪N2
degv .
The set N2of valuations is again empty, since Pv∈N2degvcan be estimated from
above by
2g−2+X
v∈V0∪N1
degv=−1.
Therefore, yi,Ψiare V0-units, and by H(Ψi) = 0 it follows that the Ψiare constants.
Finally, we consider the V0-unit equation in two variables y1+y2=1. The solutions
either satisfy
H(yi)≤2g−2+X
v∈V0∪N1∪N2
degv=−1
or they are powers of such elements. Hence the yiare also constants.
If all fractions `i(x)
`5(x)(i=1,2,3,4)
attain constant values, then by equation (4.1) also `5(x)5is a constant, as well as `5(x).
Finally, if all the linear forms
`i(x) (i=1,2,3,4,5)
attain constant values, then we get only the solutions (i, 0,0,0,0)with i=1,2,3,4 of
equation (4.1).
The calculations were carried out with KANT [2].
Acknowledgments. The authors thank the referees for valuable remarks and sugges-
tions which helped to improve the paper.
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248 István Gaál and Michael E. Pohst
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Received 30 December, 2008; revised 1 October, 2009
Author information
István Gaál, University of Debrecen, Mathematical Institute
H-4010 Debrecen Pf. 12, Hungary.
Email: [email protected]
Michael E. Pohst, Technische Universtät Berlin, Institut für Mathematik MA 8-1
Straße des 17. Juni 136, 10623 Berlin, Germany.
Email: [email protected]
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