A quantitative general Nullstellensatz for Jacobson rings
Ryota Kuroki
TL;DR
The paper addresses quantifying the general Nullstellensatz by introducing $\alpha$-Jacobson rings and an ordinal-graded proof framework. The central result is that if $A$ is $\alpha$-Jacobson, then $A[X]$ is $(\alpha+1)$-Jacobson, providing a quantitative strengthening of known cases. This yields concrete corollaries such as $K[X_1,...,X_n]$ being $(1+n)$-Jacobson for any field $K$ and $\mathbb{Z}[X_1,...,X_n]$ being $(2+n)$-Jacobson. The paper also introduces strongly Jacobson rings and shows the general Nullstellensatz extends to them, while highlighting constructive content and foundations in predicative contexts.
Abstract
The general Nullstellensatz states that if $A$ is a Jacobson ring, $A[X]$ is Jacobson. We introduce the notion of an $α$-Jacobson ring for an ordinal $α$ and prove a quantitative version of the general Nullstellensatz: if $A$ is an $α$-Jacobson ring, $A[X]$ is $(α+1)$-Jacobson. The quantitative general Nullstellensatz implies that $K[X_1,\ldots,X_n]$ is not only Jacobson but also $(1+n)$-Jacobson for any field $K$. It also implies that $\mathbb{Z}[X_1,\ldots,X_n]$ is $(2+n)$-Jacobson.