The converse of a theorem by Bayer and Stillman

Abstract

Abstract. Bayer-Stillman showed that reg(I) = reg(gin τ (I)) when τ is the graded reverse lexicographic order. We show that the reverse lexicographic order is the unique monomial order τ satisfying reg(I) = reg(gin τ (I)) for all ideals I. We also show that if gin τ1 (I) = gin τ2 (I) for all I, then τ 1 = τ 2 .