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{\bf Graham Denham}
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{\bf Short Generating Functions for some Semigroup Algebras}
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Let $a_1,\ldots,a_n$ be distinct, positive integers with
$(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let
$H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra
k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this
rational function has a simple expression in terms of $a_1,a_2,a_3$;
in particular, the numerator has at most six terms. By way of
contrast, it is known that no such expression exists for any $n\geq4$.
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