109 research outputs found
The Large Davenport Constant I: Groups with a Cyclic, Index 2 Subgroup
Let be a finite group written multiplicatively. By a sequence over ,
we mean a finite sequence of terms from which is unordered, repetition of
terms allowed, and we say that it is a product-one sequence if its terms can be
ordered so that their product is the identity element of . The small
Davenport constant is the maximal integer such that
there is a sequence over of length which has no nontrivial,
product-one subsequence. The large Davenport constant is the
maximal length of a minimal product-one sequence---this is a product-one
sequence which cannot be factored into two nontrivial, product-one
subsequences. It is easily observed that , and
if is abelian, then equality holds. However, for non-abelian groups, these
constants can differ significantly. Now suppose has a cyclic, index 2
subgroup. Then an old result of Olson and White (dating back to 1977) implies
that if is non-cyclic, and
if is cyclic. In this paper, we determine the large Davenport constant of
such groups, showing that , where is the commutator subgroup of
Zero-sum problems with congruence conditions
For a finite abelian group and a positive integer , let denote the smallest integer such that
every sequence over of length has a nonempty zero-sum
subsequence of length . We determine for all when has rank at most two and, under mild
conditions on , also obtain precise values in the case of -groups. In the
same spirit, we obtain new upper bounds for the Erd{\H o}s--Ginzburg--Ziv
constant provided that, for the -subgroups of , the Davenport
constant is bounded above by . This
generalizes former results for groups of rank two
On the arithmetic of Krull monoids with infinite cyclic class group
Let be a Krull monoid with infinite cyclic class group and let denote the set of classes containing prime divisors. We study under
which conditions on some of the main finiteness properties of
factorization theory--such as local tameness, the finiteness and rationality of
the elasticity, the structure theorem for sets of lengths, the finiteness of
the catenary degree, and the existence of monotone and of near monotone chains
of factorizations--hold in . In many cases, we derive explicit
characterizations
A new upper bound for the cross number of finite Abelian groups
In this paper, building among others on earlier works by U. Krause and C.
Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound
for the little cross number valid in the general case of arbitrary finite
Abelian groups. Given a finite Abelian group, this upper bound appears to
depend only on the rank and on the number of distinct prime divisors of the
exponent. The main theorem of this paper allows us, among other consequences,
to prove that a classical conjecture concerning the cross and little cross
numbers of finite Abelian groups holds asymptotically in at least two different
directions.Comment: 21 pages, to appear in Israel Journal of Mathematic
A nullstellensatz for sequences over F_p
Let p be a prime and let A=(a_1,...,a_l) be a sequence of nonzero elements in
F_p. In this paper, we study the set of all 0-1 solutions to the equation a_1
x_1 + ... + a_l x_l = 0. We prove that whenever l >= p, this set actually
characterizes A up to a nonzero multiplicative constant, which is no longer
true for l < p. The critical case l=p is of particular interest. In this
context, we prove that whenever l=p and A is nonconstant, the above equation
has at least p-1 minimal 0-1 solutions, thus refining a theorem of Olson. The
subcritical case l=p-1 is studied in detail also. Our approach is algebraic in
nature and relies on the Combinatorial Nullstellensatz as well as on a Vosper
type theorem.Comment: 23 page
The complete integral closure of monoids and domains II
Using geometrical methods we construct primary monoids whose complete integral closure is not completely integrally closed. Such monoids cannot be realized as multiplicative monoids of integral domains with finitely generated groups of
divisibility.
Complete integral closure, Primary monoids
The interplay of invariant theory with multiplicative ideal theory and with arithmetic combinatorics
This paper surveys and develops links between polynomial invariants of finite groups, factorization theory of Krull domains, and product-one sequences over finite groups. The goal is to gain a better understanding of the multiplicative ideal theory of invariant rings, and connections between the Noether number and the Davenport constants of finite groups. © Springer International Publishing Switzerland 2016
Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids
Arithmetical invariants---such as sets of lengths, catenary and tame
degrees---describe the non-uniqueness of factorizations in atomic monoids. We
study these arithmetical invariants by the monoid of relations and by
presentations of the involved monoids. The abstract results will be applied to
numerical monoids and to Krull monoids.Comment: 22 page
On the molecules of numerical semigroups, Puiseux monoids, and Puiseux algebras
A molecule is a nonzero non-unit element of an integral domain (resp.,
commutative cancellative monoid) having a unique factorization into
irreducibles (resp., atoms). Here we study the molecules of Puiseux monoids as
well as the molecules of their corresponding semigroup algebras, which we call
Puiseux algebras. We begin by presenting, in the context of numerical
semigroups, some results on the possible cardinalities of the sets of molecules
and the sets of reducible molecules (i.e., molecules that are not
irreducibles/atoms). Then we study the molecules in the more general context of
Puiseux monoids. We construct infinitely many non-isomorphic atomic Puiseux
monoids all whose molecules are atoms. In addition, we characterize the
molecules of Puiseux monoids generated by rationals with prime denominators.
Finally, we turn to investigate the molecules of Puiseux algebras. We provide a
characterization of the molecules of the Puiseux algebras corresponding to
root-closed Puiseux monoids. Then we use such a characterization to find an
infinite class of Puiseux algebras with infinitely many non-associated
reducible molecules.Comment: 21 pages, 2 figure
On the generalized Davenport constant and the Noether number
Known results on the generalized Davenport constant related to zero-sum
sequences over a finite abelian group are extended to the generalized Noether
number related to the rings of polynomial invariants of an arbitrary finite
group. An improved general upper bound is given on the degrees of polynomial
invariants of a non-cyclic finite group which cut out the zero vector.Comment: 14 page
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