SOME RESULTS ON ZERO-SUM PROBLEMS ARISING
IN THE THEORY OF NON-UNIQUE
FACTORIZATIONS: A SUMMARY
WOLFGANG A. SCHMID
1. Preface
The purpose of this note is to summarize several research papers,
which I, in part jointly with others, wrote and to explain the connections among them. These research papers, which are appended, in their
entirety constitute my habilitation thesis; they are referenced using the
(special) labels [1H] to [9H]. In particular, this note is not intended to
be a survey of zero-sum problems in the theory of non-unique factorizations; many important subjects are not discussed, including Davenport’s constant (and its generalizations) and the various problems
related to elasticity. Moreover, quite few references are given but some
of them to books, proceedings, and surveys where detailed accounts
and references can be found. Some of the general theory is briefly
discussed, however only with the limited scope of providing some context for the presentation of my results; in particular, occasionally only
simplified versions or special cases of results are stated.
2. Introduction
Every positive integer is in an essentially unique way the product
of prime numbers but an algebraic integer may have several essentially
different factorization into irreducible elements in the ring of integers of
an algebraic number field. A main subject of the theory of non-unique
factorizations is the description of the phenomena of non-uniqueness
of factorizations that occur in the ring of integers of algebraic number
fields and in other structures of interest, in terms of invariants of that
structure. The archetypical result of this type is that the ring of integers
of an algebraic number field is factorial if and only if it is a principal
ideal domain, i.e., its class group is trivial. The monograph [GHK06]
and the proceedings [And97, Cha05] give an extensive overview on the
state of the art and ongoing developments in this subject.
Since most of the problems concerning non-unique factorizations
turned out to be purely multiplicative, it is common and natural to
address them in a multiplicative setting, i.e., one considers (commutative, cancellative) monoids (rather than domains). An important class
of monoids that is considered in this context are Krull monoids. The
1
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
2
multiplicative monoid of the ring of integers of an algebraic number
field, and more generally any Dedekind or Krull domain, is in fact a
Krull monoid.
Many problems concerning factorizations in Krull monoids, in particular all that concern the lengths of factorizations, can be addressed
in the associated block monoid, which by definition is the monoid of all
zero-sum sequences, i.e., finite unordered sequences the sum of whose
terms equals zero, in the subset of classes containing primes of the class
group of the monoid. Thus, many questions on factorizations in a Krull
monoid can be translated into question on zero-sum sequences in an
abelian group. In this note mainly the case of finite class group will be
addressed.
The investigation of problems on zero-sum sequences has been initiated by a paper of P. Erdős, A. Ginzburg, and A. Ziv [EGZ61]. Since
that time a body of theory developed (see, e.g., [Car96] and [GHK06,
Chapter 5]) with applications to and tools from various areas of mathematics (see, e.g., [AGP94] and [Alo95, Nat96], respectively).
In the research papers that are summarized here, several problems
on zero-sum sequences in finite abelian groups that are motivated by
and can be applied to questions on factorizations are investigated. A
more detailed discussion of the content of these papers can be found
in Sections 4 and 5; now only a very brief summary of their content is
given.
• In [1H, 2H, 4H, 9H] half-factorial subsets of (various types of)
finite abelian groups are investigated with a special emphasize
on the problem of determining the maximal cardinality of a
half-factorial subset of a group and the associated inverse problem. This emphasize is due to applications of such results in
the quantitative theory of non-unique factorizations: they allow
to determine the asymptotic behavior of the classical counting
functions Gk (x) more explicitly. In particular, our results give
partial answers to a problem of W. Narkiewicz [Nar79, P 1142].
• The papers [3H, 8H] are mainly devoted to the investigation of
other quantities, denoted by ψk (G), that govern the asymptotic
behavior of Gk (x) as well. One of our results answers a problem
of J. Śliwa [Śli82, P 1247]. In these investigations various results
on half-factorial sets, in particular the inverse results mentioned
above, are applied.
• In [7H] quantities that govern the asymptotic behavior of the
counting function Bk (x) are investigated (their role is analogous
to that of ψk (G) for Gk (x)). The functions Bk (x) have been
introduced in [GHK92], they somehow lie “in between” the two
classical types of counting functions Fk (x) and Gk (x) and are
distinguished by being the “finest” block dependent counting
function of this type.
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
3
• In [6H] differences of sets of lengths are investigated and these
results are applied to the problem of arithmetically characterizing the class group of a number field (cf. [Nar04, Chapter
9]). Parts of [4H] deals with differences of sets of lengths as
well, one of the results obtained there improves a result of [6H].
On a technical level the problem of determining differences of
sets of lengths is very closely related to the investigation of
half-factorial sets: for instance, results on cross numbers of sequences are used in the investigation of both problems, and
sets that are minimal non-half-factorial play a key role in the
investigations on differences.
• In [5H] the arithmetic of block monoids is investigated from a
more general point of view than in the papers mentioned so
far, where specific arithmetical problems in block monoids are
investigated. For instance it is shown that for a finite subset
G0 of a torsion group G there exists a subset G∗0 ⊂ G that has
essentially the same arithmetic1 and in addition certain “convenient” properties. The machinery of [5H] is explicitly used in
[6H, 7H, 9H] and [Sch].
3. Basic notions of factorization theory
In this section we recall some notation and terminology that will be
needed in this summary; in particular we briefly recall in which way
problems on factorizations and on zero-sum sequences are related. The
terminology and notation follows the recent comprehensive monograph
on this subject by A. Geroldinger and F. Halter-Koch [GHK06].
Throughout, we denote by Z the integers, by N and N0 the positive and non-negative integers, respectively, and by P ⊂ N the prime
numbers; by [m, n] = {z ∈ Z : m ≤ z ≤ n} we denote the interval of
integers.
3.1. Monoids and factorizations. A monoid H is a commutative
cancellative semigroup with identity element 1H ∈ H, and multiplicative notation is used throughout. For monoids H and H 0 , a map
f : H → H 0 is called a monoid homomorphism if f (ab) = f (a)f (b)
for all a, b ∈ H. The subgroup of units (invertible elements) of H is
denoted by H × . A quotient group of H is denoted by q(H).
Two elements a, b ∈ H are called associates if there exists a unit
∈ H × such that a = b. A monoid is called reduced if its subgroup of
units is trivial. Hred = H/H × denotes the reduced monoid associated
to H.
An element a ∈ H \ H × is called an atom (or irreducible) if a has
no non-trivial divisors. An element p ∈ H \ H × is called prime if p | bc
with b, c ∈ H implies p | b or p | c. The subset of atoms and the subset
1A
transfer homomorphism from B(G0 ) to B(G∗0 ) exists.
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
4
of primes of H is denoted by A(H) and P(H), respectively. Every
prime is an atom.
The monoid H is called atomic if every a ∈ H \ H × is the product
of atoms. It is called factorial if every a ∈ H \ H × is the product of
primes. A monoid H is factorial if and only if every a ∈ H \ H × is in
an essentially unique way (cf. below for a more formal statement) the
product of atoms.
A monoid is called free if it is factorial and reduced. For a set P let
F(P ) denote the free (abelian) monoid with basis P , i.e., F(P ) is the
set of commutative formal products
(
)
Y
pvp : vp ∈ N0 and vp = 0 for almost all p ∈ P .
p∈P
P
For a = p∈P pvp ∈ F(P ), let |a| = p∈P vp denote the length of a.
Let H be an atomic monoid. The monoid Z(H) = F(A(Hred )) is
called the factorization monoid of H and the homomorphism πH :
Z(H) → Hred that is defined by “evaluating the product” is called
the factorization homomorphism of H. For a ∈ H, the set ZH (a) =
π −1 (aH × ) is called the set of factorizations of a and its elements are
called factorizations of a. The monoid H is factorial if and only if
| ZH (a)| = 1 for every a ∈ H.
For z ∈ ZH (a), |z| is called the length of the factorization and
Q
LH (a) = {|z| : z ∈ ZH (a)} ⊂ N0
is called set of lengths of a. If 0 ∈ LH (a), then a ∈ H × and LH (a) =
{0}, and if 1 ∈ LH (a), then a ∈ A(H) and LH (a) = {1}. The set
L(H) = {LH (a) : a ∈ H} is called the system of sets of length of H.
The monoid H is called half-factorial if | LH (a)| = 1 for every a ∈ H.
For L = {l1 , l2 , . . . } ⊂ N0 where li < li+i , let ∆(L) = {l2 − l1 , l3 −
lS
,
2 . . . } denote the set of (successive) distances of L. The set ∆(H) =
{∆(LH (a)) : a ∈ H} is called the set of distances of H. Clearly, the
monoid is half-factorial if and only if ∆(H) = ∅.
Let H be a monoid and F be a free monoid. A monoid homomorphism ϕ : H → F is called a divisor homomorphism if
a |H b if and only if ϕ(a) |F ϕ(b).
A divisor homomorphism ϕ : H → F is called a divisor theory if for
every f ∈ F there exists a set {a1 , . . . , an } ⊂ H such that
f = gcd({ϕ(a1 ), . . . , ϕ(an )}).
(Since F is a free monoid, every non-empty subset has a unique greatest
common divisor.)
A monoid H is called a Krull monoid if there exists a free monoid F
and a divisor homomorphism ϕ : H → F . Though, obviously not every
divisor homomorphism is a divisor theory, every Krull monoid posses
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
5
a divisor theory, which is unique up to isomorphisms. Moreover, a
monoid is a Krull monoid if and only if it is completely integrally
closed and v-noetherian (see [HK98, Chapters 22 and 23] for several
other characterization of Krull monoids). A Krull monoid is atomic
and the set of lengths of each element is finite.
Let H be a Krull monoid and ϕ : H → F a divisor theory. The
class group of H is defined as C(H) = q(F )/q(im(ϕ)). As usual, we
use additive notation for the class group. For f ∈ F we denote by [f ] ∈
C(H) the class containing f , and the subset {[p] : p ∈ P(F )} ⊂ C(H)
is referred to as the set of classes containing primes.
The notion of Krull monoids allows a unified treatment of different
structures of interest. As mentioned above, the multiplicative monoid
of a Krull domain is a Krull monoid. Since every Dedekind domain is
a Krull domain, the ring of integers of a number field as well as holomorphy rings of an algebraic function field can be investigated in this
framework. Block monoids (see below for the definition) and monoids
of certain isomorphy classes of modules under direct sum composition
(see, e.g., [FW04]) are Krull monoids as well.
3.2. Abelian groups, sequences, and block monoids. In this subsection we fix some notation for abelian groups and recall the definition of block monoids. Mainly, we will be concerned with finite abelian
groups. However, for the sake of completeness we give some definitions
and results for infinite groups as well.
Let (G, +, 0) be an abelian group. For a subset G0 ⊂ G we denote
by hG0 i the subgroup generated by G0 . A subset G0 ⊂ G \ {0} is called
independent if, for mg ∈ Z,
X
mg g = 0 implies mg g = 0 for all g ∈ G0 .
g∈G0
For a torsion element g ∈ G we denote by ord(g) ∈ N the order of the
element.
For n ∈ N, let Cn denote a cyclic group with n elements. Suppose
1 < |G| < ∞. Then there exist uniquely determined integers 1 <
n1 | . . . |nr such that
G∼
= Cn1 ⊕ · · · ⊕ Cnr ,
moreover there exist prime powers q1 , . . . , qr∗ such that G ∼
= Cq1 ⊕
∗
∗
· · · ⊕ Cqr∗ . We denote by r(G) = r the rank, by r (G) = r the total
rank, and by exp(G) = nr the exponent of G. In case |G| = 1, we set
r(G) = r∗ (G) = 0 and exp(G) = 1. The group G is called elementary if
exp(G) is squarefree and it is called a p-group if exp(G) = pk for some
p ∈ P and k ∈ N. Thus, elementary p-group means exp(G) = p ∈ P.
Let G0 ⊂ G. An element S ∈ F(G0 ) is called a sequence in G0 .
We refer to the divisors
(in F(G0 )) of S as subsequences
of S. For
Q
P
vg
a sequence S =
g∈G0 g , we denote by σ(S) =
g∈G0 vg g ∈ G
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
6
its
provided S consists only of torsion elements, by k(S) =
P sum, vand
g
g∈G0 ord(g) its cross number.
Then, | · | : F(G0 ) → N0 , σ : F(G0 ) → G, and k : F(G0 ) → Q≥0 are
monoid homomorphisms. The kernel of σ is called the block monoid
over G0 . It is denoted by B(G0 ) and its elements are called blocks
or zero-sum sequences. A sequence without a non-trivial zero-sum
subsequence is called zero-sum free.
The embedding B(G0 ) ,→ F(G0 ) is a divisor homomorphism, and
thus B(G0 ) is a Krull monoid, in particular it is atomic. The atoms
of B(G0 ) are the minimal zero-sum sequences in G0 , i.e., zero-sum
sequences such that no proper non-trivial subsequence is a zero-sum
sequence. For notational convenience, the set of atoms of B(G0 ) is
denoted just by A(G0 ) instead of A(B(G0 )), and likewise we use ∆(G0 )
and L(G0 ) to denote the sets of differences and the system of sets of
lengths, respectively, of B(G0 ).
A subset G0 ⊂ G is called half-factorial if B(G0 ) is a half-factorial
monoid. We will discuss results on half-factorial sets in detail in Section
5. In investigations on half-factorial sets in torsion groups the following
characterization due to L. Skula [Sku76] and A. Zaks [Zak76] plays a
key role: a subset G0 ⊂ G of a torsion group is half-factorial if and
only if
k(A) = 1 for every A ∈ A(G0 ).
If G is finite, then K(G) = max{k(A) : A ∈ A(G)} is called the cross
number of G and k(G) = max{k(S) : S ∈ F(G) zero-sum free} is called
the little cross number of G. The value of the [little] cross number is
only known for special types of groups, such as p-groups; however,
general upper and lower bounds are known (cf. [GHK06, Chapter 5]).
3.3. Transfer and block homomorphisms. In this subsection we
explain how questions on factorizations and zero-sum sequences are
related. An initial version of the result mentioned below, for rings of
integers, is due to W. Narkiewicz [Nar79], subsequent generalizations
are due to A. Geroldinger [Ger88] and F. Halter-Koch [HK97] (also
cf. [GHK06, Chapter 3]).
A monoid homomorphism θ : H → B is called a transfer homomorphism if
• B = θ(H)B × and θ−1 (B × ) = H × , and
• if u ∈ H, b1 , b2 ∈ B and θ(u) = b1 b2 , then there exist u1 , u2 ∈ H
such that u = u1 u2 and θ(ui ) is associated to bi for i ∈ [1, 2].
An important property of transfer homomorphisms is that they preserve sets of lengths, more formally: Let θ : H → B be a transfer
homomorphism. Then
LH (a) = LB (θ(a)),
for every a ∈ H.
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
7
Let H be a Krull monoid and ϕ : H → F(P ) a divisor theory.
Further, let G0 ⊂ C(H) denote the set of classes containing primes and
π : F(P ) → F(G0 ) the homomorphism defined by p 7→ [p].
Then β = π ◦ ϕ is called the block homomorphism of H. Since
elements of H are mapped to the zero-class, it follows that im β ⊂
B(G0 ) and indeed the following holds.
Theorem 3.1 (Narkiewicz, Geroldinger, Halter-Koch). Let H be a
Krull monoid and G0 ⊂ C(H) the set of classes containing primes.
Then
β : H → B(G0 )
is a (surjective) transfer homomorphism.
Thus, the problem of determining the sets of lengths of elements of
a Krull monoid can be transferred to the associated block monoid and
thus to a question on zero-sum sequences in abelian groups.
We point out that for the ring of integers of a number field the
class group is finite and every class contains (infinitely many) primes.
However, this is not at all true for general Krull monoids. Indeed, for
every abelian group G and for every generating (as a monoid) subset
G0 ⊂ G there exists a Krull monoid (and even a Dedekind domain)
with class group isomorphic to G such that the set of classes containing
primes corresponds to G0 ; cf. the books [Fos73, HK98].
4. Systems of sets of lengths
From the beginning of the study of non-unique factorizations the
investigation of lengths of factorizations was a main topic. For instance,
L. Carlitz [Car60] characterized number fields with class number [at
most] 2 via the property that their respective ring of integers is halffactorial [but not factorial].
It is fairly easy to see that if for a monoid H there exists some
a ∈ H whose set of lengths has more than one element, then there
exist elements in H, for instance (high) powers of a, with arbitrarily
large sets of lengths. Yet, for various classes of monoids, in particular
for Krull monoids with finite class group, it is known that (large) sets
of lengths cannot become “arbitrarily complicated,” i.e., they have a
certain structure.
Let M, d ∈ N and {0, d} ⊂ D ⊂ [0, d]. A finite non-empty set L ⊂ Z
is called and almost arithmetical multiprogression (AAMP for short)
with difference d, period D, and bound M if there exists some y ∈ Z
such that
L = y + (L0 ∪ L∗ ∪ L00 ) ⊂ y + D + dZ
with L∗ = (D + dZ) ∩ [0, max L∗ ] and −L0 , − max L∗ + L00 ⊂ [1, M ].
For an abelian group G, let
∆∗ (G) = {min ∆(G0 ) : G0 ⊂ G, ∆(G0 ) 6= ∅}.
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
8
Theorem 4.1 (Geroldinger). Let H be a Krull monoid with finite class
group G. There exist a constant MG ∈ N, just depending on G, such
that for every a ∈ H the set of lengths LH (a) is an AAMP with difference d ∈ ∆∗ (G) and bound MG .
Meanwhile, result of this type have been proved for more general
classes of monoids (see [GHK06, Chapter 4]). However, for Krull
monoids with infinite class group such a result cannot hold: by a result
of F. Kainrath [Kai99] it is known that for a Krull monoid with infinite
class group where every class contains a prime every finite set L ⊂ N≥2
is the set of lengths of some element of this monoid. Thus, one has
a complete explicit characterization of the system of sets of lengths of
such a monoid.
Of course, it would be desirable to have such an explicit characterization of the system of sets of lengths of Krull monoids with finite
class group (where every class contains a prime) as well. However,
given the complexity that can be observed in the simplest (non-trivial)
examples (see [Ger90]), such as C3 , C4 , C22 , or C23 , the goal to obtain
this in general seems very distant. Yet, in view of Theorem 4.1, a
natural step towards this goal is to investigate the set of possible differences ∆∗ (G). Given that an explicit characterization of the systems
of sets of lengths for a general finite abelian group seems out of reach,
an intermediate aim would be to “understand” them well enough to be
able to answer the following question, which was initially considered by
A. Geroldinger [Ger90]: Given some finite abelian group G, determine
all (non-isomorphic) finite abelian groups G0 such that L(G) = L(G0 ).
However, also this problem turned out to be quite a challenging one
and only partial results are known.
By L. Carlitz’s result L(C1 ) = L(C2 ) and no group with more
than 2 elements has the same system of sets of lengths. Moreover,
A. Geroldinger [Ger90] proved that L(C3 ) = L(C22 ) and no other nonisomorphic group has the same system of sets of lengths. So far no other
pair of non-isomorphic finite abelian groups with the same system of
sets of lengths is known.
There are several further results, due to A. Geroldinger [Ger90], on
this problem, in particular:
• Only finitely many non-isomorphic finite abelian groups can
have the same system of sets of lengths.
• If G is either cyclic of order at least 4 or an elementary 2-group
of rank at least 3, then no non-isomorphic group has the same
system of sets of lengths. (Analogous results are known for a
few other groups.)
Thus, it might be true that apart the two known pairs of exceptions
every finite ablian group is characterized by its system of sets of lengths.
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
9
If this should turn out to be the case it would yield a different arithmetical characterization of the class group of a number field2.
In the proof of the latter statement, results on the sets ∆∗ (G) are
applied and it is conceivable to believe that a better understanding
of the sets ∆∗ (G) would allow further progress on this problem. The
following result, which is in a similar vein and the proof of which requires knowledge on ∆∗ (G) for elementary p-groups (cf. below), might
support this believe.
Theorem 4.2 ([6H]). Let p, q ∈ P and r, s ∈ N such that r(p − 1) ≥ 3.
If L(Cpr ) = L(Cqs ), then p = q and r = s.
The condition r(p − 1) ≥ 3 is just strong enough to exclude the two
known exceptional pairs of groups. Thus, this result answers the above
mentioned question restricted to elementary p-groups (of course, this
is a massive simplification).
Before we discuss results on ∆∗ (G), we sketch how the set ∆∗ (G) is
applied to the above mentioned problems. Since the set ∆∗ (G) is so
far not known to be an invariant of G that is determined just by L(G),
one cannot apply results on ∆∗ (G) directly. However, this set is very
closely related a certain set ∆1 (G), which is determined just by L(G)
but which is difficult to investigate directly. More precisely, it is known
that ∆∗ (G) ⊂ ∆1 (G) and that every d ∈ ∆1 (G) divides some element
of ∆∗ (G). Therefore, if L(G) = L(G0 ), then max ∆∗ (G) = max ∆∗ (G0 )
and more generally all “large” elements of ∆∗ (G) and ∆∗ (G0 ) are equal.
4.1. Results on ∆∗ (G). As explained above results on max ∆∗ (G)
and “large” elements of ∆∗ (G) are of particular interest. By a result
of A. Geroldinger [Ger90] it is known that [1, r(G) − 1] ⊂ ∆∗ (G) and
that ord(g) − 2 ∈ ∆∗ (G) for each g ∈ G with ord(g) ≥ 3, in particular
(1)
max ∆∗ (G) ≥ max{exp(G) − 2, r(G) − 1}.
Equality in (1) is known to hold for several types of groups (cf. below).
In the absence of a counterexample one can hope that equality in (1)
always holds. To have such a direct dependence of max ∆∗ (G) on classical invariants of G would be particularly useful for the application we
mentioned above.
The results known so far mainly deal with groups where either the
exponent or the rank is “large” relative to the order of the group.
For instance, the first groups for which equality in (1) has been established were cyclic groups and elementary 2-groups (see [Ger90]).
It is thus known that ∆∗ (C2r ) = [1, r − 1], for cyclic groups, however, such a complete description is not known; by [GH02] it is known
that max(∆∗ (Cn ) \ {n − 2}) = bn/2c − 1. The result for elementary
2The
problem of an arithemtical characterization of the class group of a number
field was posed by W. Narkiewicz and answers were given by J. Kaczorowski and
D. Rush, cf. [Nar04, Problem 32] also see [GHK06, Chapter 7].
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
10
2-groups was considerably extended by W. Gao and A. Geroldinger
[GG00]. They proved that ∆∗ (G) = [1, r(G) − 1] if G is a p-group with
large rank3.
In [6H], see also [4H] for a refined proof, it is proved that
max ∆∗ (G) ≤ max{exp(G) − 2, 2 k(G) − 1},
which using a known upper bound on k(G) yields
max ∆∗ (G) ≤ max{exp(G) − 2, 2 log |G| − 1}.
Thus, it follows that max ∆∗ (G) = exp(G) − 2, if |G| ≤ eexp(G)/2 ,
i.e., G has a “large” expoenent. (It is also known that max ∆∗ (G) =
exp(G) − 2 if |G| ≤ exp(G)2 , see [GG00] and also [GHK06, Section
6.8].) If G is an elementary 2-group, then 2 k(G) − 1 = r(G) − 1 and
the bound given above is sharp as well. However, this is a rather
exceptional case4, since the size of k(G) is at least p−1
r∗ (G) where p
p
is the smallest prime divisor of |G|. Moreover, it is proved that for
elementary p-groups equality always holds in (1), more precisely it is
shown that for p ∈ P and r ∈ N
[1, r − 1] ∪ [max{1, p − r − 1}, p − 2] ⊂ ∆∗ (Cpr ) ⊂
[1, r − 1] ∪ [max{1, p − r − 1}, p − 2] ∪ [1, (p − 3)/2],
in particular ∆∗ (Cpr ), and ∆1 (Cpr ) as well, is an interval if and only if
p ≤ 2r + 1.
We end this section with a remark on the proofs of the results of
[6H]. By definition of ∆∗ (G) one has to investigate min ∆(G0 ) for
subsets G0 ⊂ G that are not half-factorial. There are several results
known that allow to restrict ones attention to sets G0 with additional
properties. For instance, a simple but very useful observation is that if
one is only interested in upper bounds for max ∆∗ (G), then it suffices
to investigate minimal non-half-factorial sets5. Another “reduction”
result that was applied is the one obtained in [5H] that is mentioned in
Section 2. Moreover, there are several results on min ∆(G0 ) for subsets
G0 ⊂ G with additional properties. For instance, if there exists some
A ∈ A(G0 ) with cross number less than 1, then min ∆(G0 ) ≤ exp(G)−2
(see [GG00]). Another type of set that plays an important role are
“simple” sets; these are sets that consist of independent elements and
one additional element and fulfil a certain minimality constraint (see
[5H]). In a nutshell, to prove results on ∆∗ (G) one first restricts to
investigating the sets G0 ⊂ G that are “interesting” and then divides
3More
precisely, this holds if r(G) ≥ (exp(G) − 1) + (exp(G) − 1)2 (exp(G) −
2)/2. More generally, for G an arbitrary finite abelian groups it is known that
max ∆∗ (G) ≤ r∗ (G) − 1 if r∗ (G) ≥ (exp(G) − 1) + (exp(G) − 1)2 (exp(G) − 2)/2.
4Indeed, it is the only case if equality should always holds in (1).
5A set is called minimal non-half-factorial if it is not half-factorial but every
proper subset is half-factorial.
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
11
the “interesting” sets into certain subcategories each of which is then
investigated separately.
5. Quantitative theory of non-unique factorizations
In the 1960s W. Narkiewicz started a systematic investigation of
quantitative aspects of phenomena of non-unique factorization in number fields6, i.e., problems of the following type: Let H denote the set
of non-zero principal ideals of R, the ring of integers of some number
field7. For a subset Z ⊂ H that is defined via some property P related
to non-unique factorizations (for instance, the set of principal ideals
that have at most 2 essentially different factorizations in H) determine
the asymptotic behavior of the function Z(x) that is defined as the
number of elements in Z with (absolute) norm at most x.
These question have been a major impetus for the development of
the general theory sketched in Section 3 and there are many contributions to this subject, in particular (in roughly chronological order) by
W. Narkiewicz, P. Rémond, J. Śliwa, J. Kaczorowski, A. Geroldinger,
and F. Halter-Koch; we refer to the books [Nar04, Chapter 9] and
[GHK06, Chapters 8 and 9] for a presentation of this subject and its
development.
It turned out that the order of magnitude8 of Z(x), for an ample
variety of properties P, including all that we consider here, is given by
x(log x)−AP (log log x)BP ,
where AP and BP just depend on the class group of R. Moreover,
the (numerical) value for the constants AP and BP can be obtained
by solving certain combinatorial problems in the class group of R. For
several properties P these problems can be conveniently formulated in
the associated block monoid. In the following two subsections results on
the constants AP and BP for two (classical) properties P are discussed.
5.1. Invariants related to Gk (x) and half-factorial sets. Let K
be a number field with class group G and ring of integers R. Further,
let H denote the monoid of non-zero principal ideals of R. For k ∈ N
let Gk (H) = {a ∈ H : | LH (a)| ≤ k} and Gk (x) = Gk (H)(x) = |{a ∈
Gk (H) : |a| ≤ x}|, where |a| = (R : a) denotes the (absolute) norm of a.
6Meanwhile,
an abstract setting has been developed that allows, in particular,
to treat number fields and function fields simultaneously (cf. [GHK06, Chapters 8
and 9]).
7Equivalently, one considers (non-zero) elements of R, but then only considers
the number of non-associated elements.
8By a result of J. Kaczorowski [Kac83] much more precise asymptotics are known.
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
12
It is known that if9 |G| ≥ 3, then
(2)
Gk (x) x(log x)−1+µ(G)/|G| (log log x)ψk (G)
where µ(G) and ψk (G) are non-negative integers that depend just on
G, and G and k, respectively. Their precise combinatorial description
is given below. Let Gk (G) = {B ∈ B(G) : | L(B)| ≤ k}. For G0 ⊂ G
and S ∈ F(G \ G0 ), let Ω(G0 , S) = S · F(G0 ) ∩ B(G). Then
µ(G) = max{|G0 | : G0 ⊂ G half-factorial}
and
ψk (G) = max{|S| : G0 ⊂ G half-factorial with |G0 | = µ(G) and
S ∈ F(G \ G0 ) with ∅ =
6 Ω(G0 , S) ⊂ Gk (G)}.
For a proof of (2) in present terminology see [GHK06, Section 9.4].
W. Narkiewicz [Nar79, P 1142] posed the problem of evaluating
µ(G); in view of the “definition” of ψk (G), the associated inverse problem is of interest as well. So far it has been solved only in special cases.
Several of my papers deal with this problem.
In [1H] A. Plagne and I, building on the work of A. Geroldinger
and J. Kaczorowski [GK92], determined µ(G) for elementary p-groups.
Moreover, we solved the associated inverse problem, i.e., we determined
the structure of half-factorial subsets with cardinality (close to) µ(G)
in elementary p-groups. Prior to that this was only known for exponent
2, 3, 5, and 7: the result for exponent 2 is due to W. Narkiewicz [Nar79],
M. Radziejewski [Rad02] obtained the result for exponent 3, and in [9H]
it is given for exponent 5 and 7, as well as for 3 (it is slightly different
to M. Radziejewski’s and was obtained independently). However, for
these groups with small exponent more is known, namely, the structure
of all half-factorial sets (not only of those with “large” cardinality). It
turns out that for these groups each indecomposable component of a
half-factorial set is contained in a subgroup of rank at most exp(G)−1.
(“Indecomposable component” means that the block monoid over this
set is not the direct product of two proper submonoids; a detailed
discussion of this notion can be found in [5H].) It might be interesting
to note that for groups of exponent 4 a result of this type cannot hold
(see [3H]).
In [2H] A. Plagne and I investigated µ(G) for cyclic groups. We
showed that µ(Cn ) ≥ 1 + τ (n)/2, for n ≥ 2, where τ (n) denotes the
number of divisors of n. In combination with the well known bound
µ(Cn ) ≤ τ (n), see [Śli82], this yields the order of magnitude of µ(Cn ).
Moreover, we showed that µ(Cn ) = τ (n) if n is a product of two prime
powers (this was known already to be true for prime powers [Sku76,
Śli76, Zak76] and the product of two primes [GK92]). However, by
9By
Carlitz’s result the case |G| ≤ 2 is a degenerate one, which we do not include,
for convenience and clarity, in the present discussion.
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
13
an example due to A. Zaks [Zak80], it is known that µ(Cn ) = τ (n)
does not always hold and we showed that this gap can be arbitrarily
large; more precisely, we can construct sequences (ni )i∈N such that
τ (ni ) − µ(ni ) log τ (ni ) and τ (ni ) → ∞. Moreover, we studied µ(Cn )
for n the product of three distinct primes: µ(Cn ) is either 7 or 8, and
both values occur infinitely often (more precisely, we obtained some
criteria, depending on congruence relations among the primes, that
allow to determine the precise value in various cases). We did so as
well for n the product of four distinct primes and obtained similar,
though somewhat weaker, results.
In [4H] M. Radziejewski and I investigated weakly half-factorial (whf
for short) sets. A subset G0 ⊂ G of a finite abelian group is called whf
if k(A) ∈ N for each A ∈ A(G0 ). Whf sets were introduced by J. Śliwa
[Śli82] and further investigated by W. Gao and A. Geroldinger [GG98].
Using a known characterization of whf sets via characters of G, we
gave an explicit description of all (inclusion maximal) whf sets, and in
particular we explicitly determine µ0 (G), the maximal cardinality of a
whf subset of G. Since every half-factorial set is whf, we thus obtain
an improved general upper bound for µ(G).
Moreover, combining the result on whf sets with known results on
the cross number of finite abelian groups, we obtain µ(G) easily for
all groups with total rank at most two and with additional effort for
some other groups. Though, it is very likely that some further insight
on half-factorial sets can be obtained using the result on whf sets, it
should be noted that with growing (total) rank the connection between
whf and half-factorial sets becomes increasingly weak (among others10
we proved that if µ(G) = µ0 (G), then G has rank at most two).
In [3H], jointly with M. Radziejewski, and in [8H] the constant ψk (G)
is investigated. Motivated by a recent result of M. Radziejewski [Rad05]
on oscillations of the error-term of Gk (x), we tried to show, in [3H], that
ψk (G) is positive11. We established the positivity for k ≥ 2; this can be
done (almost) without any knowledge on half-factorial sets. For k = 1,
which is obviously the hardest case, we proved the positivity for various classes of groups, including all those for which at the time of the
writing of the paper µ(G) was known. It seems (to me) that in order to
prove ψ1 (G) > 0 one needs some knowledge on a half-factorial subset
with maximal cardinality of G; however, quite incomplete knowledge
on this half-factorial set can actually be sufficient, not even its cardinality µ(G) needs to be known. In particular, we proved ψ1 (G) > 0
“only” assuming that a half-factorial subset of G with maximal cardinality fulfils some condition. We did so for several different conditions,
all of which are actually known to hold for certain classes of groups. In
10The
actual result is slightly technical.
ψk (G) is positive, then a result of M. Radziejewski implies that Gk (x)
“oscillates.”
11If
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
14
fact, many of the results we obtained for specific classes of groups where
obtained by observing (or asserting) that these classes are covered by
our “abstract” results. In [8H] the aim was a different one, namely
to obtain the precise value of or at least good bounds for ψk (G). In
view of the definition of ψk (G) it is apparent that this requires detailed
knowledge on the half-factorial subsets with maximal cardinality of G.
Most results are concerned with elementary p-groups with even rank,
where these sets are well understood: for these groups the precise value
of ψ1 (G) is obtained and that of ψk (G), for arbitrary k, provided the
exponent of the group is not smaller than the rank. For elementary
2-groups (regardless whether the rank is even or odd) the problem of
determining ψk (G) was shown to be equivalent to a problem in extremal graph theory, namely the problem of determining the maximal
number of edges in a graph on r vertices, where r corresponds to the
rank of G, without k edge-disjoint cycles. This equivalence and results on the graph theoretic problem (cf. [Bol78, Chapter III]) imply
that (ψk (G))k∈N is not always an arithmetic progression, which answers
[Śli82, P 1247] to the negative12.
5.2. Invariants related to Bk (x). Again, let K be a number field
with class group G and ring of integers R, and let H denote the
monoid of non-zero principal ideals of R. For k ∈ N let Bk (H) = {a ∈
H : | Z(β(a))| ≤ k} and Bk (x) = Bk (H)(x) = |{a ∈ Bk (H) : |a| ≤ x}|,
where |a| = (R : a) denotes the (absolute) norm of a.
This counting function has been introduced by A. Geroldinger and
F. Halter-Koch [GHK92]. There it is shown that if13 |G| ≥ 3, then
(3)
Bk (x) x(log x)−1+(1+r
∗ (G))/|G|
(log log x)bk (G) ,
where
bk (G) = max{|S| : G0 ⊂ G factorial, |G0 | = 1 + r∗ (G) and
S ∈ F(G \ G0 ) with ∅ =
6 Ω(G0 , S) ⊂ Bk (G)}.
A set is called factorial if the block monoid over this set is factorial. By
a result of [GHK92] a subset G0 ⊂ G is factorial if and only if G0 \ {0}
is independent. Thus, 1+r∗ (G) is the maximal cardinality of a factorial
subset of G0 .
In [7H] several results on bk (G) were obtained. The precise value of
bk (G) for elementary groups is determined: Let G ∼
= Cp1 ⊕ · · · ⊕ Cpr
with primes pi , then
r
X
r2 (G)
,
bk (G) = (k − 1) max{pi : i ∈ [1, r]} +
(pi − 1) −
2
i=1
more explicit investigation shows that already for G ∼
= C24 the sequence
(ψk (G))k∈N is not an arithmetic progression, which allows to give explicit examples
of number fields with that property.
13As for G (x) this is no actual restriction and only imposed for convenience.
k
12A
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
15
where r2 (G) denotes the 2-rank of G, i.e., equality holds at the lower
bound (implicitly) obtained in [GHK92]. If G is not elementary, then
not every factorial set with maximal cardinality in G is a generating set
of G, which makes determining bk (G) more difficult. Using an addition
theorem due to I. Chowla (cf. [Nat96, Chapter 2]), the precise value of
bk (G) is determined in case G is a cyclic group of prime power order. A
combination of these two results was then used to determine bk (G) for a
further class of groups. Moreover, two complementary lower bounds for
b1 (G), for general G, are obtained: One “extends” the result for cyclic
groups of prime power order and, naturally, yields for these groups as
well as for elementary groups the precise value. The other one, which
is Davenport’s constant of G modulo a minimal essential subgroup of
G, though it is trivial for elementary groups yields considerably better
bounds than the former for various types of groups. For instance if
n = pk q l with k, l ≥ 2 and distinct primes p and q, then the former
yields b1 (Cn ) ≥ pk + pk−1 + q l + q l−1 − 4 whereas the latter yields
b1 (Cn ) ≥ pk−1 q l−1 .
References
[AGP94] W. R. Alford, A. Granville, and C. Pomerance. There are infinitely many
Carmichael numbers. Ann. of Math. (2), 139(3):703–722, 1994.
[Alo95] N. Alon. Tools from higher algebra. In Handbook of combinatorics, Vol.
1, 2, pages 1749–1783. Elsevier, Amsterdam, 1995.
[And97] D. D. Anderson, editor. Factorization in integral domains, volume 189 of
Lecture Notes in Pure and Applied Mathematics, New York, 1997. Marcel
Dekker Inc.
[Bol78] B. Bollobás. Extremal graph theory, volume 11 of London Mathematical
Society Monographs. Academic Press Inc., London, 1978.
[Car60] L. Carlitz. A characterization of algebraic number fields with class number
two. Proc. Amer. Math. Soc., 11:391–392, 1960.
[Car96] Y. Caro. Zero-sum problems—a survey. Discrete Math., 152(1-3):93–113,
1996.
[Cha05] S. T. Chapman, editor. Arithmetical properties of commutative rings and
monoids, volume 241 of Lecture Notes in Pure and Applied Mathematics.
Chapman & Hall/CRC, Boca Raton, FL, 2005.
[EGZ61] P. Erdős, A. Ginzburg, and A. Ziv. Theorem in the additive number
theory. Bull. Res. Council Israel, 10F:41–43, 1961.
[Fos73] R. M. Fossum. The divisor class group of a Krull domain. SpringerVerlag, New York, 1973.
[FW04] A. Facchini and R. Wiegand. Direct-sum decompositions of modules with
semilocal endomorphism rings. J. Algebra, 274(2):689–707, 2004.
[Ger88] A. Geroldinger. Über nicht-eindeutige Zerlegungen in irreduzible Elemente. Math. Z., 197(4):505–529, 1988.
[Ger90] A. Geroldinger. Systeme von Längenmengen. Abh. Math. Sem. Univ.
Hamburg, 60:115–130, 1990.
[GG98] W. Gao and A. Geroldinger. Half-factorial domains and half-factorial
subsets of abelian groups. Houston J. Math., 24(4):593–611, 1998.
[GG00] W. Gao and A. Geroldinger. Systems of sets of lengths. II. Abh. Math.
Sem. Univ. Hamburg, 70:31–49, 2000.
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
[GH02]
16
A. Geroldinger and Y. ould Hamidoune. Zero-sumfree sequences in cyclic
groups and some arithmetical application. J. Théor. Nombres Bordeaux,
14(1):221–239, 2002.
[GHK92] A. Geroldinger and F. Halter-Koch. Nonunique factorizations in block
semigroups and arithmetical applications. Math. Slovaca, 42(5):641–661,
1992.
[GHK06] A. Geroldinger and F. Halter-Koch. Non-unique factorizations. Algebraic,
Combinatorial and Analytic Theory. Chapman & Hall/CRC, 2006.
[GK92] A. Geroldinger and J. Kaczorowski. Analytic and arithmetic theory of
semigroups with divisor theory. Sém. Théor. Nombres Bordeaux (2),
4(2):199–238, 1992.
[HK97] F. Halter-Koch. Finitely generated monoids, finitely primary monoids,
and factorization properties of integral domains. In [And97] pages 31–72.
[HK98] F. Halter-Koch. Ideal systems, volume 211 of Monographs and Textbooks
in Pure and Applied Mathematics. Marcel Dekker Inc., New York, 1998.
[Kac83] J. Kaczorowski. Some remarks on factorization in algebraic number fields.
Acta Arith., 43(1):53–68, 1983.
[Kai99] F. Kainrath. Factorization in Krull monoids with infinite class group.
Colloq. Math., 80(1):23–30, 1999.
[Nar79] W. Narkiewicz. Finite abelian groups and factorization problems. Colloq.
Math., 42:319–330, 1979.
[Nar04] W. Narkiewicz. Elementary and analytic theory of algebraic numbers.
Springer-Verlag, Berlin, third edition, 2004.
[Nat96] M. B. Nathanson. Additive number theory, volume 165 of Graduate Texts
in Mathematics. Springer-Verlag, New York, 1996.
[1H]
A. Plagne and W. A. Schmid. On large half-factorial sets in elementary
p-groups: Maximal cardinality and structural characterization. Israel J.
Math., 145:285–310, 2005.
[2H]
A. Plagne and W. A. Schmid. On the maximal cardinality of half-factorial
sets in cyclic groups. Math. Ann., 333(4):759–785, 2005.
[Rad02] M. Radziejewski. Wybrane zagadnienia teorii funkcji L wraz z zastosowaniami. PhD Thesis, Adam Mickiewicz University, Poznań, 2002.
[Rad05] M. Radziejewski. On the distribution of algebraic numbers with prescribed factorization properties. Acta. Arith., 116:153–171, 2005.
[3H]
M. Radziejewski and W. A. Schmid. On the asymptotic behavior of some
counting functions. Colloq. Math., 102(2):181–195, 2005.
[4H]
M. Radziejewski and W. A. Schmid. Weakly half-factorial sets in finite
abelian groups. Forum Math., to appear.
[5H]
W. A. Schmid. Arithmetic of block monoids. Math. Slovaca, 54(5):503–
526, 2004.
[6H]
W. A. Schmid. Differences in sets of lengths of Krull monoids with finite
class group. J. Théor. Nombres Bordeaux, 17(1):323–345, 2005.
[7H]
W. A. Schmid. On invariants related to non-unique factorizations in block
monoids and rings of integers. Math. Slovaca, 55(1):21–37, 2005.
[8H]
W. A. Schmid. On the asymptotic behavior of some counting functions,
II. Colloq. Math., 102(2):197–216, 2005.
[9H]
W. A. Schmid. Half-factorial sets in elementary p-groups. Far East J.
Math. Sci. (FJMS), to appear.
[Sch]
W. A. Schmid. Periods of sets of lengths: a quantitative result and an
associated inverse problem, submitted.
[Sku76] L. Skula. On c-semigroups. Acta Arith., 31(3):247–257, 1976.
SOME RESULTS ON ZERO-SUM PROBLEMS: A SUMMARY
[Śli76]
[Śli82]
[Zak76]
[Zak80]
17
J. Śliwa. Factorizations of distinct lengths in algebraic number fields. Acta
Arith., 31(4):399–417, 1976.
J. Śliwa. Remarks on factorizations in algebraic number fields. Colloq.
Math., 46(1):123–130, 1982.
A. Zaks. Half factorial domains. Bull. Amer. Math. Soc., 82(5):721–723,
1976.
A. Zaks. Half-factorial-domains. Israel J. Math., 37(4):281–302, 1980.
Institut für Mathematik und wissenschaftliches Rechnen, KarlFranzenzens-Universität Graz, Heinrichstraße 36, 8010 Graz, Austria
E-mail address: [email protected]