$S$-Integral Points in Orbits on $\mathbb{P}^1$

TL;DR

The paper studies the number of $S$-integral points in forward orbits on $\\mathbb{P}^1$ under a rational map $\\varphi$ of degree $d\\ge 2$ over a number field, with a non-preperiodic $\\alpha$ and a non-exceptional $\\beta$. Using a single-place quantitative Roth approach together with height machinery, it proves a sharp upper bound of $c_1|S|(\\log|S|+1)^5$ on the count of $n$ for which $\\varphi^n(\\alpha)$ is $S$-integral relative to $\\beta$, uniformly in $\\beta$ and independent of the orbit details. The authors also establish uniform bounds in the special case where $\\varphi$ is a polynomial with a uniformly bounded number of places of bad reduction, by leveraging Looper’s uniform height results and quasi-$(S,\\epsilon)$-integrality to handle conjugations. The work extends prior exponential bounds from HS11 and KLS15, and clarifies when uniformity can be achieved within dynamical Siegel-type finiteness questions on $\\mathbb{P}^1$.

Abstract

Let $K$ be a number field and $S$ a finite set of places of $K$ that contains all of the archimedean places. Let $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ be a rational map of degree $d \geq 2$ defined over $K$. Given $α\in \mathbb{P}^1(K)$ non-preperiodic and $β\in \mathbb{P}^1(K)$ non-exceptional, we prove an upper bound of the form $O(|S|^{1+ε})$ on the number of points in the forward orbit of $α$ that are $S$-integral relative to $β$, extending results of Hsia--Silverman [HS11]. We also prove uniform bounds when $\varphi$ is a polynomial, extending resaults of Krieger et al [KLS+15].

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• proof
• Theorem 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• proof