A Gorenstein numerical semi-group ring having a transcendental series of Betti numbers
A numerical semigroup \(S\) is an additive semigroup of natural numbers generated by a set of numbers \(0<g_1<g_2<\dots<g_n\) such that \(gcd(g_1,\dots,g_n)=1\). Let \(k\) be a field of characteristic zero. The numerical semigroup ring \(k[S]\) of \(S\) over \(k\) is by definition the subring of the polynomial ring \(k[t]\) generated by the monomials \(t^{g_1},t^{g_2},\dots,t^{g_n}\). The Poincaré-Betti series of \(k[S]\) is by definition the generating series \(P_{k[S]}(z)=\sum_{i\geq 0}\mid\mathrm{Tor}_i^{k[s]}(k,k)\mid z^i\). According to \textit{J. Tate} [Ill. J. Math. 1, 14--27 (1957; Zbl 0079.05501)] a complete intersection always has a rational Poincaré-Betti series. In the paper under review, the authors prove that there are semigroups \(S\) such that \(G=k[S]\) is Gorenstein ring and such that \(P_G(z)\) is a transcendental function. More precisely, let \(R\) is the subring of the polynomial ring \(k[t]\), generated by the twelve monomials \(t^{36},t^{48},t^{50},t^{52},t^{56},t^{60},t^{66},t^{67},t^{107},t^{121},t^{129},t^{135}\). Then \(R\) is a Gorenstein ring and the generating series \(P_R(z)\) is rationally related to the infinite product \(\prod_{n=1}^{\infty}\frac{(1+z^{2n-1})^2}{(1-z^{2n})^2}\) and thus transcendental.
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