Arithmetic Nullstellensatz and Nonstandard Methods

TL;DR

This paper establishes height bounds for polynomial rings over various domains using nonstandard methods, with applications to primality testing and factorization, despite the bounds being ineffective.

Contribution

It introduces nonstandard techniques to derive height bounds in polynomial rings over integral domains and algebraic numbers, extending to valuation rings and arithmetical functions.

Findings

Height bounds for polynomial rings over integral domains.

Application of bounds to primality testing of ideals.

Analysis of bounds in UFDs and valuation rings.

Abstract

In this study we find height bounds for polynomial rings over integral domains. We apply nonstandard methods and hence our constants will be ineffective. Then we find height bounds in the polynomial ring over algebraic numbers to test primality of an ideal. Furthermore we consider unique factorization domains and possible bounds for valuation rings and arithmetical functions.