Strong Nœther Position and Stabilized Regularities

Abstract

In this article, we introduce the notion of a homogeneous ideal in strong Nœther position (SNP); a new definition for the notion of generic coordinates for some problems. This definition is simple to check, because it can be tested on the initial ideal for the degree reverse lexicographic ordering. It is explicit, because we provide an algorithm to decide whether a monomial ideal is in SNP or not. To any notion of regularity of an ideal (Castelnuovo–Mumford regularity, satiety, maximal degree of the elements of the reduced Gröbner basis with respect to the degree reverse lexicographic ordering, and Hilbert regularity), we associate a stabilized regularity. The main result of this article is that SNP shows simply and explicitly the relationships between these regularities.

Key Words:

• Castelnuovo–Mumford regularity
• Generic coordinates
• Gröbner basis
• Hilbert regularity
• Satiety

2000 Mathematics Subject Classification:

• Primary 13F20
• Secondary 13P10

ACKNOWLEDGMENTS

I would like to thank Prof. Daniel Lazard for suggesting to me the idea of this work and for his very helpful comments. This work was supported in part by the CEAMA, Isfahan, University of Technology, Isfahan, Iran.

Notes

Communicated by I. Swanson.