Pre-images of quadratic dynamical systems

TL;DR

This paper investigates bounds on the number of rational pre-images of constants under quadratic dynamical systems over the rationals, providing explicit bounds for algebraic constants and exploring the geometry of pre-image surfaces.

Contribution

It extends previous work by establishing explicit bounds for all algebraic constants and analyzing the geometry of pre-image surfaces in quadratic dynamical systems.

Findings

Bounded the number of rational pre-images for algebraic constants

Analyzed the geometry of pre-image surfaces

Provided explicit bounds for specific constants like zero and -1

Abstract

For a quadratic endomorphism of the affine line defined over the rationals we consider the problem of bounding the number of rational points that eventually land at a given constant after iteration, called pre-images of the constant. In the article "Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems," it was shown that the number of rational pre-images is bounded as one varies the morphism in a certain one-dimensional family. Explicit values of the constant for pre-images of zero and -1 defined over the rational numbers were addressed in subsequent articles. This article addresses an explicit bound for any algebraic image constant and provides insight into the geometry of the "pre-image surfaces."