Stretched Gorenstein Rings

The Poincaré series of a local Gorenstein ring of multiplicity up to 10 is rational

“…We can thus consider the local, Artinian ring A := R/J and its maximal ideal M := N /J . With the above notations, following Sally (see [Sa1]), we say that a ring R is stretched if length(M 2 / M 3 ) = 1. In [Sa2] the rationality of P R is proved when R is a stretched Cohen-Macaulay ring.…”

On the depth of the tangent cone and the growth of the Hilbert function

“…In [Abh67] Abhyankar proved that if e 0 = e 0 (R) is the multiplicity of R and b = dim k (m/m 2 ) is its embedding dimension, then we have Sally in a long series of papers studied the depth of the tangent cone in terms of the difference ε = e 0 − (b − d + 1) ≥ 0; see [Sal77], [Sal79a], [Sal79c], [Sal80a], [Sal80b], and [Sal83]. She proved that if ε = 0 then G(m) is Cohen-Macaulay, [Sal77], and proposed the following conjecture, [ This conjecture was proved by Rossi and Valla,[RosV96a], and independently by Wang, [Wan97].…”

A family of local rings with rational Poincaré series

“…Instead, if the square of the maximal ideal of an Artinian reduction is minimally generated by two elements, we say that A is almost stretched. See [11], [9], [5] and [6] for papers concerning these notions.…”