785 research outputs found
Cauchy-Davenport type theorems for semigroups
Let be a (possibly non-commutative) semigroup. For we define , where is the set of the units of , and The paper
investigates some properties of and shows the following
extension of the Cauchy-Davenport theorem: If is cancellative and
, then This
implies a generalization of Kemperman's inequality for torsion-free groups and
strengthens another extension of the Cauchy-Davenport theorem, where
is a group and in the above is replaced by the
infimum of as ranges over the non-trivial subgroups of
(Hamidoune-K\'arolyi theorem).Comment: To appear in Mathematika (12 pages, no figures; the paper is a sequel
of arXiv:1210.4203v4; shortened comments and proofs in Sections 3 and 4;
refined the statement of Conjecture 6 and added a note in proof at the end of
Section 6 to mention that the conjecture is true at least in another
non-trivial case
Alon's Nullstellensatz for multisets
Alon's combinatorial Nullstellensatz (Theorem 1.1 from \cite{Alon1}) is one
of the most powerful algebraic tools in combinatorics, with a diverse array of
applications. Let \F be a field, be finite nonempty
subsets of \F. Alon's theorem is a specialized, precise version of the
Hilbertsche Nullstellensatz for the ideal of all polynomial functions vanishing
on the set S=S_1\times S_2\times ... \times S_n\subseteq \F^n. From this Alon
deduces a simple and amazingly widely applicable nonvanishing criterion
(Theorem 1.2 in \cite{Alon1}). It provides a sufficient condition for a
polynomial which guarantees that is not identically zero
on the set . In this paper we extend these two results from sets of points
to multisets. We give two different proofs of the generalized nonvanishing
theorem. We extend some of the known applications of the original nonvanishing
theorem to a setting allowing multiplicities, including the theorem of Alon and
F\"uredi on the hyperplane coverings of discrete cubes.Comment: Submitted to the journal Combinatorica of the J\'anos Bolyai
Mathematical Society on August 5, 201
On the critical pair theory in abelian groups : Beyond Chowla's Theorem
We obtain critical pair theorems for subsets S and T of an abelian group such
that |S+T| < |S|+|T|+1. We generalize some results of Chowla, Vosper, Kemperman
and a more recent result due to Rodseth and one of the authors.Comment: Submitted to Combinatorica, 23 pages, revised versio
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