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Some arithmetic aspects of polynomial maps

Wodson Mendson

Abstract

The Jacobian conjecture is a well-known open problem in affine algebraic geometry that asks if any polynomial endomorphism of the affine space $\mathbb{A}_{\mathbb{C}}^{n}$ ($n\geq2$) with jacobian $1$ is an automorphism. We present a survey about some results around this conjecture and we discuss an arithmetic aspect of this conjecture due to Essen-Lipton. We investigate some cases of this arithmetic approach showing the close relationship between the Jacobian Conjecture and the problem of counting $\mathbb{F}_p$-points of an affine scheme.

Some arithmetic aspects of polynomial maps

Abstract

The Jacobian conjecture is a well-known open problem in affine algebraic geometry that asks if any polynomial endomorphism of the affine space () with jacobian is an automorphism. We present a survey about some results around this conjecture and we discuss an arithmetic aspect of this conjecture due to Essen-Lipton. We investigate some cases of this arithmetic approach showing the close relationship between the Jacobian Conjecture and the problem of counting -points of an affine scheme.

Paper Structure

This paper contains 8 sections, 35 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Notation
  4. Polynomial maps
  5. Unimodular and invariant domains
  6. Invariance and unimodularity
  7. $\mathbb{F}_p$-points of hypersurfaces and unimodularity
  8. Keller-finite domains

Key Result

Theorem 1

The $p$-adic integer ring $\mathbb{Z}_{p}$ does satisfy the Unimodular Conjecture for almost all primes $p$ if and only if the Jacobian Conjecture is true.

Theorems & Definitions (82)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Conjecture
  • Proposition 1
  • proof
  • Theorem 3
  • proof
  • Proposition 2
  • proof
  • ...and 72 more