On the Hilbert function of certain non Cohen-Macaulay one-dimensional rings
Let \(k\) be a field, and \(n_0,\dots,n_r\) positive integers which have greatest common divisor 1. Denote by \(A\) the ring \(k[[t^{n_0}, \dots, t^{n_r}]]\), by \(M\) its maximal ideal, and by \(G=\text{gr}_M (A)=\bigoplus_{i\geq 0}(M^i/M^{i+1})\) the associated graded ring of \(A\) with respect to \(M\). The Hilbert series \[ H_G(z)=\sum_{i\geq 0} \dim_k(M^i/M^{i+1}) z^i \] is of the form \(Q_G(z)/(1-z)\) where \(Q_G(z)\in \mathbb{Z}[z]\), \(Q_G(z)=\sum^s_{i=0} h_iz^i\), \(h_s\neq 0\). If \(G\) is Cohen-Macaulay, then it is well-known that \(h_i>0\) for \(i=0, \dots,s\). The authors study some of the coefficients \(h_i\) in the case in which \(G\) is not necessarily Cohen-Macaulay. They prove that \(h_s>0\) in any case and that \(h_2\geq 0\), \(h_3\geq 0\) under suitable assumptions.
scientific article; zbMATH DE number 1309620