Reduced Gr\"obner Bases of Certain Toric Varieties; A New Short Proof

TL;DR

This paper presents a simplified proof for the minimal Gröbner basis of a toric ideal associated with certain affine toric varieties defined by almost arithmetic sequences, correcting previous inaccuracies.

Contribution

It provides a new, shorter proof for the minimal Gröbner basis of specific toric ideals and establishes necessary and sufficient conditions for this basis to be reduced.

Findings

Corrects previous work on Gröbner bases of toric ideals

Provides a minimal Gröbner basis for a class of toric varieties

Establishes conditions for the basis to be reduced

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 paper we produce a minimal Gr\"obner 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 Gr\"obner basis of C, correcting a previous work of \cite{Sen} and giving a much simpler proof than that of \cite{Ayy}.