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Maximal Representations of Elements in Numerical Semigroups
Maximal Representations of Elements in Numerical Semigroups
2011-02-27
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arxiv.org/abs/1102.5518v1
Mathematics
Commutative Algebra
Scientific paper
Given a numerical semigroup $S = $ and $s\in S$, we consider the representations $s = c_1 a_1 + c_2 a_2 +...+ c_t a_t$ where $c_i\ge0$. Such a representation is {\em maximal} if $c_1+c_2+...+c_t$ is a maximum over all such representations of $s$. We show that the number of maximal representations, varying over the elements in $S$, is always bounded. Thus, we define $\mp(S)$ to be the maximum number of maximal representations of elements in $S$. We study maximal representations in depth when $S$ has embedding dimension 3, and establish a formula for $\mp(S)$ in this case.
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