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Let $(R, \mathfrak{m}) = k[[X^a, X^b, X^c]]$, $a<b<c$, $gcd(a, b, c) = 1$, be a semigroup ring. We have $R$ is a Cohen-Macaulay local ring of dimension one. It is well known that $\ell(R/\mathfrak{m}^{n+1})$ becomes a polynomial for large enough $n$ (in fact $n \ge a$) we have $$\ell(R/\mathfrak{m}^{n+1}) = e_0 (n+1) - e_1,$$ where $e_0$ (resp. $e_1$) is called the multiplicity (resp. the Chern number) of $R$. It is not difficult to see that $e_0 = a$.

Question. What is the complexity of the computation $e_1$?

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