PreprintarXiv licencearXiv
Degree Growth of Iterates of Curves and Likely Intersections
Abstract
We study the growth of the bidegree of an ample irreducible curve in ℙ1 × ℙ1 under a product polynomial endomorphism φ=(f,g), where at least one of f and g is non-exceptional. We prove that, if the curve C is not preperiodic under (fa,gb) for any a,b≥ 1, then the bidegree of φn(C) is asymptotic to (deg(g)n,deg(f)n). As an application of this exponential growth, we prove a geometric analogue of Silverman's theorem on the finiteness of S-integral points in orbits. Namely if C is not (fa,gb)-preperiodic and C' is not totally invariant for φ, then for any infinite sequence ni of positive integers, the union of the intersections ⋃i ≥ 1 ( φni(C)∩ C' ) is Zariski dense in C'.
Related
- Likely and unlikely ramification of algebraic points on curves
- Triple intersections for algebraic curves on the real torus
- Intersections of fractal sets with rectifiable curves
- Beyond Multi-Index Models: Learning Intersections of Degree-2 PTFs
- On the non-existence of certain maximizing curves of degree 7
BibTeX
@misc{saleh2026degree,
title = {{Degree Growth of Iterates of Curves and Likely Intersections}},
author = {Sina Saleh and Jit Wu Yap},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20580},
eprint = {2609.20580},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20580},
}
RIS
TY - UNPB
TI - Degree Growth of Iterates of Curves and Likely Intersections
AU - Sina Saleh
AU - Jit Wu Yap
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.20580
UR - https://arxiv.org/abs/2609.20580
AB - We study the growth of the bidegree of an ample irreducible curve in ℙ^1 × ℙ^1 under a product polynomial endomorphism φ=(f,g), where at least one of f and g is non-exceptional. We prove that, if the curve C is not preperiodic under (f^a,g^b) for any a,b≥ 1, then the bidegree of φ^n(C) is asymptotic to (deg(g)^n,deg(f)^n). As an application of this exponential growth, we prove a geometric analogue of Silverman's theorem on the finiteness of S-integral points in orbits. Namely if C is not (f^a,g^b)-preperiodic and C' is not totally invariant for φ, then for any infinite sequence n_i of positive integers, the union of the intersections ⋃_i ≥ 1 ( φ^n_i(C)∩ C' ) is Zariski dense in C'.
ER -