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Acta Arithmetica

The catenary degree of Krull monoids I

Journal de Théorie des Nombres de Bordeaux

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf{c}\left(H\right)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each two factorizations $z,{z}^{\prime }$ of $a$, there exist factorizations $z={z}_{0},...,{z}_{k}={z}^{\prime }$ of $a$ such that, for each $i\in \left[1,k\right]$, ${z}_{i}$ arises from ${z}_{i-1}$ by replacing at most $N$ atoms from ${z}_{i-1}$ by at most $N$ new atoms. Under a very mild condition...

Representation of finite abelian group elements by subsequence sums

Journal de Théorie des Nombres de Bordeaux

Let $G\cong {C}_{{n}_{1}}\oplus ...\oplus {C}_{{n}_{r}}$ be a finite and nontrivial abelian group with ${n}_{1}|{n}_{2}|...|{n}_{r}$. A conjecture of Hamidoune says that if $W={w}_{1}·...·{w}_{n}$ is a sequence of integers, all but at most one relatively prime to $|G|$, and $S$ is a sequence over $G$ with $|S|\ge |W|+|G|-1\ge |G|+1$, the maximum multiplicity of $S$ at most $|W|$, and $\sigma \left(W\right)\equiv 0\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}|G|$, then there exists a nontrivial subgroup $H$ such that every element $g\in H$ can be represented as a weighted subsequence sum of the form $g=\underset{i=1}{\sum ^{n}}{w}_{i}{s}_{i}$, with ${s}_{1}·...·{s}_{n}$ a subsequence of $S$. We give two examples showing this does not hold in general, and characterize the counterexamples...

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