Preimages of quadratic dynamical systems
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Publication:444576
DOI10.2140/INVOLVE.2011.4.343zbMATH Open1258.37076arXiv1007.0744OpenAlexW2565347413MaRDI QIDQ444576
Trevor Hyde, Benjamin Krause, Benjamin Hutz
Publication date: 15 August 2012
Published in: Involve (Search for Journal in Brave)
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."
Full work available at URL: https://arxiv.org/abs/1007.0744
rational pointsarithmetic dynamicsarithmetic geometryuniform boundpreimagequadratic dynamical systems
Rational points (14G05) Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets (37F10) Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps (37P05)
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