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On multiplicatively dependent vectors of polynomial values

Marley Young

Abstract

Given polynomials $f_1,\ldots,f_n$ in $m$ variables with integral coefficients, we give upper bounds for the number of integral $m$-tuples $\mathbf{u}_1,\ldots, \mathbf{u}_n$ of bounded height such that $f_1(\mathbf{u}_1), \ldots, f_n(\mathbf{u}_n)$ are multiplicatively dependent. We also prove, under certain conditions, a finiteness result for $\mathbf{u} \in \mathbb{Z}^m$ with relatively prime entries such that $f_1(\mathbf{u}),\ldots,f_n(\mathbf{u})$ are multiplicatively dependent.

On multiplicatively dependent vectors of polynomial values

Abstract

Given polynomials in variables with integral coefficients, we give upper bounds for the number of integral -tuples of bounded height such that are multiplicatively dependent. We also prove, under certain conditions, a finiteness result for with relatively prime entries such that are multiplicatively dependent.
Paper Structure (10 sections, 11 theorems, 46 equations)

This paper contains 10 sections, 11 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Conventions and notation
  3. Statement of results
  4. Acknowledgements
  5. A finiteness result
  6. Statistical results
  7. Weil height
  8. Counting integral solutions to polynomial equations
  9. Multiplicative structure of algebraic numbers
  10. Proofs of Propositions \ref{['prop:NaiveCount']} and \ref{['prop:RankCount']}, and Theorem \ref{['thm:AICount']}

Key Result

Theorem 1.1

OSSZ Let $F=(f_1,\ldots,f_n) \in K(X)^n$, whose components cannot multiplicatively generate a power of a linear fractional transformation. Then there are only finitely many elements $\alpha \in K^{\mathrm{ab}}$ such that $F(\alpha)$ is multiplicatively dependent.

Theorems & Definitions (22)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Definition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Theorem 1.8
  • Proposition 1.9
  • Proposition 2.1
  • ...and 12 more