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Counting rational maps on $\mathbb{P}^1$ with prescribed local conditions

Khoa D. Nguyen, Anwesh Ray

TL;DR

This work addresses counting rational maps on $\mathbb{P}^1$ over $\mathbb{Q}$ with prescribed local reductions by introducing and studying the global minimal resultant. It develops a density framework combining height densities and weak box densities, and proves positive-density results for families of maps with globally minimal resultant, including explicit degree-2 computations yielding $>32.7\%$ squarefree resultants. The approach blends Schanuel-type height counts, Schwartz–Zippel bounds on resultants, sieve arguments for local conditions, and computational algebra (Macaulay2) to analyze local structures. The findings illuminate how integral models and good reduction behave in arithmetic dynamics, providing a parallel to Cremona–Sadek-type density results for elliptic curves and offering quantitative insight into reductions of dynamical systems over $\mathbb{Q}$.

Abstract

We explore distribution questions for rational maps on the projective line $\mathbb{P}^1$ over $\mathbb{Q}$ within the framework of arithmetic dynamics, drawing analogies to elliptic curves. Specifically, we investigate counting problems for rational maps $φ$ of fixed degree $d \geq 2$ with prescribed reduction properties. Our main result establishes that the set of rational maps with minimal resultant has positive density. Additionally, for degree 2 rational maps, we perform explicit computations demonstrating that over $32.7\%$ possess a squarefree, and hence minimal, resultant.

Counting rational maps on $\mathbb{P}^1$ with prescribed local conditions

TL;DR

This work addresses counting rational maps on over with prescribed local reductions by introducing and studying the global minimal resultant. It develops a density framework combining height densities and weak box densities, and proves positive-density results for families of maps with globally minimal resultant, including explicit degree-2 computations yielding squarefree resultants. The approach blends Schanuel-type height counts, Schwartz–Zippel bounds on resultants, sieve arguments for local conditions, and computational algebra (Macaulay2) to analyze local structures. The findings illuminate how integral models and good reduction behave in arithmetic dynamics, providing a parallel to Cremona–Sadek-type density results for elliptic curves and offering quantitative insight into reductions of dynamical systems over .

Abstract

We explore distribution questions for rational maps on the projective line over within the framework of arithmetic dynamics, drawing analogies to elliptic curves. Specifically, we investigate counting problems for rational maps of fixed degree with prescribed reduction properties. Our main result establishes that the set of rational maps with minimal resultant has positive density. Additionally, for degree 2 rational maps, we perform explicit computations demonstrating that over possess a squarefree, and hence minimal, resultant.
Paper Structure (16 sections, 27 theorems, 143 equations)

This paper contains 16 sections, 27 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Methodology
  5. Acknowledgements
  6. The height density, box density, and weak box densities
  7. The height density
  8. The box density and weak box densities
  9. Projective sets with prescribed local conditions and their affine liftings
  10. Local conditions and affine liftings
  11. Some admissibility conditions
  12. Minimal resultants
  13. The minimal resultant
  14. Equations over finite fields
  15. Some positive densities
  16. ...and 1 more sections

Key Result

Theorem A

Let $d\geq 3$ be an odd integer and let $\Sigma$ be a finite set of prime numbers. Let $\mathcal{W}_\Sigma$ be the set of rational maps $\phi(z)\in\mathbb{Q}(z)$ of degree $d$ with $2d$-power free resultant and good reduction at all primes $p\in \Sigma$. The following assertions hold: Consequently, with respect to the height density, a positive proportion of rational maps have good reduction at a

Theorems & Definitions (63)

  • Theorem A: Remark \ref{['rem:V and W have globally minimal maps']} and Theorem \ref{['main thm of section 4 Wsigma']}
  • Theorem B: Remark \ref{['rem:V and W have globally minimal maps']} and Theorem \ref{['main thm of section 4']}
  • Theorem C: Theorem \ref{['d=2 main thm']}
  • Definition 2.1
  • Proposition 2.4: Schanuel
  • proof
  • Lemma 2.5: Schwartz-Zippel
  • proof
  • Proposition 2.6
  • proof
  • ...and 53 more