Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantitative bounds on integrality for post-critically finite maps

Rudranarayan Padhy, Sudhansu Sekhar Rout

Abstract

Let $K$ be a number field with algebraic closure $\overline{K}$ and let $S$ be a finite set of places of $K$ that contain all the archimedean places. For an integer $d \ge 2$, consider the unicritical polynomial family $f_{d,c}(z) = z^d + c$. Recently, Benedetto and Ih studied the distribution of post-critically finite parameters $c$ that are $S$-integral relative to a fixed point $α\in \overline{K}$ such that $f_{d, α}$ is not post-critically finite under some additional assumptions. In this paper we study the quantitative aspects of their result. In particular, we establish quantitative bounds on the number of $S$-integral post-critically finite parameters in the generalized Mandelbrot set $\mathcal{M}_{d, v}$ relative to a non post-critically finite parameter $α$ with some additional assumptions.

Quantitative bounds on integrality for post-critically finite maps

Abstract

Let be a number field with algebraic closure and let be a finite set of places of that contain all the archimedean places. For an integer , consider the unicritical polynomial family . Recently, Benedetto and Ih studied the distribution of post-critically finite parameters that are -integral relative to a fixed point such that is not post-critically finite under some additional assumptions. In this paper we study the quantitative aspects of their result. In particular, we establish quantitative bounds on the number of -integral post-critically finite parameters in the generalized Mandelbrot set relative to a non post-critically finite parameter with some additional assumptions.
Paper Structure (10 sections, 16 theorems, 98 equations)

This paper contains 10 sections, 16 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Canonical heights
  4. Bifurcation measure
  5. The Arakelov--Zhang pairing
  6. Equidistribution of post-critically finite points
  7. Linear forms in logarithms
  8. Proof of Main Results
  9. Proof of Theorem \ref{['thm2']}
  10. Proof of Theorem \ref{['thm02']}

Key Result

Theorem 1.2

Suppose that for every archimedean place $v$ of $K$ and for every $K$-embeddings $\tau:K(\alpha)\to \mathbb{C}_v$, the image $\tau(\alpha)$ does not lie in the boundary $\partial \mathcal{M}_{d,v}$ of the $v$-adic generalized Mandelbrot set $\mathcal{M}_{d,v}$. Then Conjecture conj1 holds.

Theorems & Definitions (24)

  • Conjecture 1.1
  • Theorem 1.2: Benedetto
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 14 more