Nullstellens\"atze and Applications

TL;DR

This paper provides straightforward proofs of key Nullstellensatz theorems and explores diverse applications across algebra, topology, and combinatorics, demonstrating their broad utility in solving classical problems.

Contribution

It offers simplified proofs of major Nullstellensatz results and applies them to solve longstanding problems in various mathematical fields.

Findings

Simplified proofs of Classical, Real, Projective, and Combinatorial Nullstellensatz.

Applications include solutions to Hilbert's 17th problem and the Borsuk-Ulam theorem.

Construction of a non-Euclidean principal ideal domain.

Abstract

In this expository paper, we present simple proofs of the Classical, Real, Projective and Combinatorial Nullstellens\"atze. Several applications are also presented such as a classical theorem of Stickelberger for solutions of polynomial equations in terms of eigenvalues of commuting operators, construction of a principal ideal domain which is not Euclidean, Hilbert's $17^{th}$ problem, the Borsuk-Ulam theorem in topology and solutions of the conjectures of Dyson, Erd\"{o}s and Heilbronn.