The converse of a theorem by Bayer and Stillman

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 $gin_{τ_1}(I) = gin_{τ_2}(I)$ for all $I$, then $τ_1 = τ_2$.

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