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On a problem posed by Bjorn Poonen

James Rawson

Abstract

Bjorn Poonen asked whether there exists a polynomial giving a surjection $\mathbb{Z} \times \mathbb{Z} \to \mathbb{N}$. We answer this question in the negative, conditional on a conjecture of Vojta. More precisely, we show that if such a function exists, there is a family of open surfaces with dense integral points despite the surfaces being of log general type.

On a problem posed by Bjorn Poonen

Abstract

Bjorn Poonen asked whether there exists a polynomial giving a surjection . We answer this question in the negative, conditional on a conjecture of Vojta. More precisely, we show that if such a function exists, there is a family of open surfaces with dense integral points despite the surfaces being of log general type.
Paper Structure (5 sections, 17 theorems, 1 figure)

This paper contains 5 sections, 17 theorems, 1 figure.

Table of Contents

  1. Integral Points Are Not Dense on $X_n$
  2. Geometry of $X_n$ and Complements of Plane Curves
  3. Arithmetic of Generically Rational Polynomials
  4. Case 1:
  5. Case 2:

Key Result

Theorem 1

Suppose integral points are not dense on $X_n$ for some odd integer $n$, then $F(\mathbb{Z}, \mathbb{Z}) \neq \mathbb{N}$.

Figures (1)

  • Figure 1: An illustration of the region used in case 1 of the proof of Theorem \ref{['case2a']}.

Theorems & Definitions (31)

  • Theorem 1
  • Conjecture 1: Vojta
  • Conjecture 2: Bombieri-Lang
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 21 more