The structure of generic subintegrality (Q1901336)

Let \(x_1, \dots, x_p\), \(z\) be independent indeterminates over \(\mathbb{Q}\). Set \(x_0 = 1\) and \(\gamma_n = \sum^p_{i = 0} {n \choose i} x_i z^{n - i}\). The authors study the ring \(R^{(N)} = \mathbb{Q} [\{\gamma_n \mid n \geq N\}]\) graded by weight\((x_i) = i\), weight\((z) = 1 \). The motivation is that \(z\) is subintegral over \(R^{(N)}\) and the extension \(R^{(N)} \subseteq \mathbb{Q} [x_1, \dots, x_p,z]\) is universal for subintegral elements. The ideal of relations among the \(\gamma_n\)'s is generated by quadratic relations. Use of a basis interchange technique to produce a \(\mathbb{Q}\)-basis for \(R^{(N)}\) then gives the Poincaré series \[ \sum^\infty_{k = 0} t^k \dim R_k^{(N)} = (1 - t + t^{N + p})/(1 - t) (1 - t^N) (1 - t^{N + 1}) \dots (1 - t^{N + p - 1}). \] This generalises certain results of \textit{L. G. Roberts} and \textit{B. Singh} [J. Pure Appl. Algebra 95, No. 3, 331-351 (1994; see the preceding review)]. It yields the Hilbert function of \(R^{(N)}\) in terms of \(d\) polynomials where \(d = l\text{ cm} (N,N + 1, \dots, N + p - 1)\). Furthermore, \(R^{(N)}\) is Cohen-Macaulay of Krull dimension \(p + 1\), and a minimal presentation of \(R^{(N)}\) is obtained with \(N + 2p\) generators and \({N + p \choose p}\) relations. The following conditions are equivalent: (i) \(R^{(N)}\) is Gorenstein, (ii) \(R^{(N)}\) is a complete intersection, (iii) \(N + p < 2\).