Short Generating Functions for some Semigroup Algebras
Keyword(s):
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 $n\geq4$.
Keyword(s):
2019 ◽
Vol 15
(05)
◽
pp. 1037-1050
1981 ◽
Vol 81
(1)
◽
pp. 19-19
◽
2019 ◽
Vol 101
(1)
◽
pp. 35-39
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Keyword(s):
Keyword(s):