Reduced Gröbner Bases of Certain Toric Varieties; A New Short Proof

Abstract

Let K be a field and let m ₀,…,m _n be an almost arithmetic sequence of positive integers. Let C be a toric variety in the affine (n + 1)-space, defined parametrically by x ₀ = t ^{m ₀} ,…,x _n  = t ^{m _n} . In this article we produce a minimal Gröbner basis for the toric ideal which is the defining ideal of C and give sufficient and necessary conditions for this basis to be the reduced Gröbner basis of C, correcting a previous work of [6] and giving a much simpler proof than that of [1].

Key Words:

• Gröbner basis
• Monomial curves
• Polynomial ideals
• Toric varieties

2000 Mathematics Subject Classification:

• 13P10

ACKNOWLEDGMENTS

The author thanks Professor Irena Swanson for the useful discussions and comments during the course of this work. Also, the author thanks the referee for the very useful suggestion that simplified the proof much easier than the original form.

The author thanks the referee for suggesting to use a result of [2] that shortened the proof.

Notes

Communicated by W. Bruns.