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The catenary degree of Krull monoids I

Alfred GeroldingerDavid J. GrynkiewiczWolfgang A. Schmid — 2011

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 c ( H ) of H is the smallest integer N with the following property: for each a H and each two factorizations z , z of a , there exist factorizations z = z 0 , ... , z k = z of a such that, for each i [ 1 , k ] , 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

David J. GrynkiewiczLuz E. MarchanOscar Ordaz — 2009

Journal de Théorie des Nombres de Bordeaux

Let G C n 1 ... 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 | | W | + | G | - 1 | G | + 1 , the maximum multiplicity of S at most | W | , and σ ( W ) 0 mod | G | , then there exists a nontrivial subgroup H such that every element g H can be represented as a weighted subsequence sum of the form g = n i = 1 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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