Counting Polynomial-type Exceptional Units on Algebraic Varieties over Number Fields

TL;DR

This paper extends the study of polynomial-type exceptional units to algebraic number fields and varieties, providing exact formulas and asymptotic estimates for their counts, especially improving bounds for low-degree varieties.

Contribution

It introduces a new framework for counting polynomial-type exceptional units over algebraic number fields and derives explicit formulas and asymptotic bounds using advanced number theoretic techniques.

Findings

Derived an exact counting formula for exceptional units on varieties with good reduction.

Established asymptotic estimates for the number of exceptional units.

Improved error bounds for varieties of degree at most two.

Abstract

Previous research on exceptional units has primarily focused on the ring of rational integers or abstract finite rings, often restricted to linear or quadratic constraints. In this paper, we extend the concept of polynomial-type exceptional units to the ring of integers of an arbitrary algebraic number field. We investigate the number of these polynomial-type exceptional units on general algebraic varieties. By employing the Chinese Remainder Theorem and Hensel's lifting technique, we derive an exact counting formula for the number of these exceptional units on a smooth closed subscheme under the assumption of good reduction. Furthermore, using the Lang-Weil inequality, we establish an asymptotic estimate for the counting function. In particular, we prove that for varieties of degree at most two, the error term can be significantly improved, yielding a sharper asymptotic bound.