Pre-images of quadratic dynamical systems

Details Pre-images of quadratic dynamical systems Pre-images of quadratic dynamical systems

2010-07-05

arxiv.org/abs/1007.0744v2

Mathematics

Number Theory

to appear in Involve; 16pages

Scientific paper

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."

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