Infinite Versions of Hilbert's Nullstellensatz

TL;DR

The paper addresses the nullstellensatz in polynomial rings of infinite Krull dimension by collecting and proving a wide network of equivalent formulations. It establishes a detailed, elementary chain of implications among conditions, including a geometric description of maximal ideals, and proves that the base field must be algebraically closed with a cardinality constraint $|I| < |F|$ for the equivalences to hold. A central contribution is a persistence result showing the strong Nullstellensatz holds in extensions to larger polynomial rings when certain cardinality conditions are met. The work also provides explicit constructions and tricks (e.g., Lang’s construction and Rabinowitsch’s trick) to connect algebraic and geometric viewpoints, with corollaries relating radical ideals, vanishing sets, and extensions.

Abstract

We compile a long list of equivalent formulations of Hilbert's Nullstellensatz in infinite dimensions, and prove a persistence result for the strong Nullstellensatz in large polynomial rings.

Theorems & Definitions (8)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Corollary 1.5
• Definition 2.1
• Proposition 2.2
• proof : Proof of Corollary \ref{['cor:idealfrombelow']}