Effective Genericity, δ-Regularity and Strong Noether Position

Abstract

We show that the concept of strong Noether position for a polynomial ideal ℐ ▵ left𝒫 is equivalent to δ-regularity and thus related to Pommaret bases. In particular, we provide explicit Pommaret bases for two of the ideal sequences used in Hashemi's definition of strong Noether position and alternative proofs for a number of his statements. Finally, we show that one consequence of δ-regularity is that any Pommaret basis contains a system of parameters and we present an algorithm for checking whether the factor ring 𝒫/ℐ is Gorenstein via a socle computation.

Key Words:

• Castelnuovo-Mumford regularity
• Hilbert regularity
• Pommaret basis
• Quasi-stable ideal
• Satiety

2010 Mathematics Subject Classification:

• Primary 13P10
• Secondary 13F20, 14Q20, 68W30

Notes

¹In the literature, further names for the same class of monomial ideals appear like ideals of Borel type [14] or weakly stable ideals [3].

This is the only point where we need the assumption that 𝕜 is an infinite field. In the case of a small finite field, the construction of δ-regular variables might require a field extension.

Later, quasi-regularity was rediscovered by Schenzel et al. [17] under the name filter-regularity. It is amusing to note that in the same letter Serre already described the Eisenbud-Goto criterion for q-regularity (found in 1984 [5]) as a “curiosité”.

In [22] it was falsely claimed that the equality follows immediately from Theorem 2.8. However, as one can see in our proof, this argument yields only an inequality. Now we close this gap by also proving the converse inequality.

In the context of differential equations an explicit expression is e.g., given in [19].

We exploit here that in the homogeneous case ℐ and lt ℐ always have the same dimension. The proposition remains true for non-homogeneous ideals, if we restrict to degree compatible orders.

Communicated by I. Swanson.