The converse of a theorem by Bayer and Stillman

Abstract

Bayer-Stillman showed that $reg(I)=reg(gi{n}_{\tau }(I))$ when $\tau$ is the graded reverse lexicographic order. We show that the reverse lexicographic order is the unique monomial order $\tau$ satisfying $reg(I)=reg(gi{n}_{\tau }(I))$ for all ideals $I$. We also show that if $gi{n}_{{\tau }_1}(I)=gi{n}_{{\tau }_2}(I)$ for all $I$, then ${\tau }_1={\tau }_2$.

arXiv comment10 pages