A quantitative general Nullstellensatz for Jacobson rings

TL;DR

This paper introduces a quantitative version of the Nullstellensatz for Jacobson rings, defining $eta$-Jacobson rings and proving that polynomial extensions increase this property by one, with specific results for fields and integers.

Contribution

It defines $eta$-Jacobson rings and proves a quantitative Nullstellensatz, showing how Jacobson properties extend in polynomial rings with explicit bounds.

Findings

$A[X]$ is $(eta+1)$-Jacobson if $A$ is $eta$-Jacobson.

$K[X_1,...,X_n]$ is $(1+n)$-Jacobson for any field $K$.

$bZ[X_1,...,X_n]$ is $(2+n)$-Jacobson.

Abstract

The general Nullstellensatz states that if $A$ is a Jacobson ring, $A[X]$ is Jacobson. We introduce the notion of an $\alpha$-Jacobson ring for an ordinal $\alpha$ and prove a quantitative version of the general Nullstellensatz: if $A$ is an $\alpha$-Jacobson ring, $A[X]$ is $(\alpha+1)$-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.