On Noether differential operators attached to a zero-dimensional primary ideal – shape basis case.

*(English)*Zbl 1140.32302
Kazama, Hideaki (ed.) et al., Proceedings of the 12th international conference on finite or infinite dimensional complex analysis and applications, ICFIDCAA, Tokyo, Japan, July 27– 31, 2004. Fukuoka: Kyushu University Press (ISBN 4-87378-899-4/pbk). 357-366 (2005).

Let \(I\) be a zero-dimensional primary ideal in the polynomial ring in \(n\) variables \(K[x] = K[x_1, x_2, \dots, x_n]\), over the rational field \(K = \mathbb{Q}\). Then Grothendieck duality shows that the vector space \(\mathop{\mathrm{Ext}}^n_{K[x]}(K[x]/I, K[x])\) is the dual space of \(K[x]/I\) and each element of \(\mathop{\mathrm{Ext}}^n_{K[x]}(K[x]/I, K[x])\) can be regarded as a linear functional or a generalised function acting on \(K[x]\) that annihilates \(I\). Since this pairing can be described in terms of Grothendieck local residues, the generalized functions in \(\mathop{\mathrm{Ext}}^n_{K[x]}(K[x]/I, K[x])\) act on \(K[x]\) as linear partial differential operators.

In earlier work, the author has shown that these operators can be expressed in terms of what he refers to as Noether differential operators attached to \(I\). The ideal \(I\) is said to have a shape basis if it has a basis of the form \(\{g_1(x_1), x_2-g_2(x_1), \dots, x_n-g_n(x_1)\}\) where \(g_1, \dots, g_n\in K[x_1]\). In this paper, the author studies the Noether differential operators attached to \(I\) when \(I\) has such a basis. In particular he presents a GrĂ¶bner basis algorithm for computing these operators.

For the entire collection see [Zbl 1105.00004].

In earlier work, the author has shown that these operators can be expressed in terms of what he refers to as Noether differential operators attached to \(I\). The ideal \(I\) is said to have a shape basis if it has a basis of the form \(\{g_1(x_1), x_2-g_2(x_1), \dots, x_n-g_n(x_1)\}\) where \(g_1, \dots, g_n\in K[x_1]\). In this paper, the author studies the Noether differential operators attached to \(I\) when \(I\) has such a basis. In particular he presents a GrĂ¶bner basis algorithm for computing these operators.

For the entire collection see [Zbl 1105.00004].

Reviewer: John Clark (Dunedin)