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

Chen Lin, Kaihan Tang

TL;DR

The paper extends the concept of polynomial-type exceptional units (f-exunits) to the ring of integers $\mathcal{O}_K$ of a number field and defines $\mathscr{E}_{f,\mathfrak{n}}(X)$ for smooth closed subschemes $X\subset \mathbb{A}^n_{\mathcal{O}_K}$. It establishes an explicit exact counting formula for $\#\mathscr{E}_{f,\mathfrak{n}}(X)$ under good reduction, built from a multiplicative Euler product derived via the Chinese Remainder Theorem and a Hensel lifting argument for prime powers. The paper further provides Lang-Weil-based asymptotic estimates for the counting function, with a sharpened bound when $\deg X\le 2$, reflecting improved error terms. The methods combine local-to-global lifting, local point counts, and global geometric input to quantify polynomial-type units on varieties over number fields, offering tools that generalize prior integer- and finite-ring results and suggesting directions for singular fibers and function-field analogues.

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.

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

TL;DR

The paper extends the concept of polynomial-type exceptional units (f-exunits) to the ring of integers of a number field and defines for smooth closed subschemes . It establishes an explicit exact counting formula for under good reduction, built from a multiplicative Euler product derived via the Chinese Remainder Theorem and a Hensel lifting argument for prime powers. The paper further provides Lang-Weil-based asymptotic estimates for the counting function, with a sharpened bound when , reflecting improved error terms. The methods combine local-to-global lifting, local point counts, and global geometric input to quantify polynomial-type units on varieties over number fields, offering tools that generalize prior integer- and finite-ring results and suggesting directions for singular fibers and function-field analogues.

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.
Paper Structure (4 sections, 7 theorems, 34 equations)

This paper contains 4 sections, 7 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Asymptotic estimates for the counting function
  4. Concluding Remarks

Key Result

Theorem 1.2

Given a number field $K$ with ring of integers $\mathcal{O}_K$, let $f\in \mathcal{O}_K[x]$ be a polynomial, and $X$ be a smooth closed subscheme of $\mathbb{A}^n_{\mathcal{O}_K}$ with $\mathrm{codim}(X)=d$. For a nonzero integral ideal $\mathfrak{n}$ of $\mathcal{O}_K$ such that $X$ has a good redu where $\mathcal{N}^f(\mathfrak{p},X)=\{(x_1,\dots,x_n)\in X(\mathcal{O}_K/\mathfrak{p}):\prod_{i=1}

Theorems & Definitions (16)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 6 more