Determination of all rational preperiodic points for morphisms of PN

For a morphism $f:¶^N \to ¶^N$, the points whose forward orbit by $f$ is finite are called preperiodic points for $f$. This article presents an algorithm to effectively determine all the rational preperiodic points for $f$ defined over a given number field $K$. This algorithm is implemented in the open-source software Sage for $\Q$. Additionally, the notion of a dynatomic zero-cycle is generalized to preperiodic points. Along with examining their basic properties, these generalized dynatomic cycles are shown to be effective.

• Publication year

  2012

• Venue

  Mathematics of Computation

• Publication date

  2012-10-23

• Fields of study

  Mathematics, Computer Science

• Identifiers
  DOI 10.1090/S0025-5718-2014-02850-0arXiv 1210.6246
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• On a class of monic binomials

  2013cited by this paper
• Computing algebraic numbers of bounded height

  2011cited by this paper
• Effectivity of Dynatomic cycles for morphisms of projective varieties using deformation theory

  2010cited by this paper
• Short Bases of Lattices over Number Fields

  2010cited by this paper
• On Poonen's Conjecture Concerning Rational Preperiodic Points of Quadratic Maps

  2009cited by this paper
• On Commutative Algebra

  2009cited by this paper
• Good reduction of periodic points on projective varieties

  2009cited by this paper
• Computing points of small height for cubic polynomials

  2008cited by this paper
• Rational 6-Cycles Under Iteration of Quadratic Polynomials

  2008cited by this paper
• Good Reduction of Periodic Points

  2008cited by this paper
• Dynatomic cycles for morphisms of projective varieties

  2008cited by this paper
• Finding Rational Periodic Points on Wehler K3 Surfaces

  2008cited by this paper
• ℚ‐rational cycles for degree‐2 rational maps having an automorphism

  2008cited by this paper
• A Computational Investigation of Wehler K3 Surfaces

  2008cited by this paper
• Rational Periodic Points for Degree Two Polynomial Morphisms on Projective Space

  2008cited by this paper
• The Arithmetic of Dynamical Systems

  2007cited by this paper
• SAGE: system for algebra and geometry experimentation

  2005cited by this paper
• Thesis

  2002cited by this paper
• Algorithmic Number Theory

  2000cited by this paper
• The classification of rational preperiodic points of quadratic polynomials over ${\Bbb Q}$: a refined conjecture

  1998cited by this paper
• On certain algebraic curves related to polynomial maps

  1996cited by this paper
• Bornes pour la torsion des courbes elliptiques sur les corps de nombres

  1996cited by this paper
• Commutative Algebra: with a View Toward Algebraic Geometry

  1995cited by this paper
• The Complete Classification of Rational Preperiodic Points of Quadratic Polynomials over Q: A Refined Conjecture

  1995cited by this paper
• Cycles of quadratic polynomials and rational points on a genus-$2$ curve

  1995cited by this paper
• Periodic points, multiplicities, and dynamical units.

  1995cited by this paper
• Rational periodic points of rational functions

  1994cited by this paper
• Several Complex Variables and the Geometry of Real Hypersurfaces

  1993cited by this paper
• Sur quelques problèmes de dynamique holomorphe

  1992cited by this paper
• Factoring polynomials with rational coefficients

  1982cited by this paper
• Excess intersection of divisors

  1981cited by this paper
• Algèbre linéaire sur $K[X_1,\dots,X_n]$ et élimination

  1977cited by this paper
• The Algebraic Theory of Modular Systems

  1972cited by this paper
• Periodic Points on an Algebraic Variety

  1950cited by this paper
• Numerical Evidence for a Conjecture of Poonen

  year unknowncited by this paper

• Counting the number of $m$-periodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, V

  2025influential citation
• Counting the number of $n$-periodic integral points of a discrete dynamical system with applications from arithmetic statistics, IV

  2025influential citation
• Counting the number of $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-fixed points of a discrete dynamical system with applications from arithmetic statistics, III

  2025influential citation
• Counting the number of $2$-periodic $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI

  2025influential citation
• Counting the number of $\mathcal{O}_{K}$-fixed points of a discrete dynamical system with applications from arithmetic statistics, II

  2025influential citation
• Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I

  2024cites this paper
• Portraits of quadratic rational maps with a small critical cycle

  2024influential citation
• On the proportion of irreducible polynomials in unicritically generated semigroups

  2023cites this paper
• Quadratic points on dynamical modular curves

  2023cites this paper
• Dynatomic polynomials, necklace operators, and universal relations for dynamical units

  2021cites this paper
• N T ] 8 M ay 2 02 1 Rational Periodic Points of x d + c and abc Conjecture

  2021influential citation
• Rational Periodic Points of xd + c and Fermat-Catalan Equations

  2021influential citation
• Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line

  2020influential citation
• SMALLEST REPRESENTATIVES OF SL(2,Z)-ORBITS OF BINARY FORMS AND ENDOMORPHISMS OF P

  2018cites this paper
• Smallest representatives of $\rm {SL}(2,\mathbb {Z})$-orbits of binary forms and endomorphisms of $\mathbb {P}^1$

  2018cites this paper
• Preperiodic points for quadratic polynomials over cyclotomic quadratic fields

  2018cites this paper
• Dynamical modular curves for quadratic polynomial maps

  2017cites this paper
• SAGE as a Source for Undergraduate Research Projects

  2017cites this paper
• Symmetrization of rational maps: Arithmetic properties and families of Lattès maps of ℙ^{𝕜}

  2016cites this paper
• Preperiodic portraits for unicritical polynomials

  2015cites this paper
• Quadratic maps with a periodic critical point of period 2

  2015influential citation
• Automorphism groups and invariant theory on ℙN

  2015cites this paper
• Dynamics of quadratic polynomials over quadratic fields

  2014cites this paper
• Arithmetic dynamics on smooth cubic surfaces

  2013cites this paper
• Preperiodic points for quadratic polynomials over quadratic fields

  2013cites this paper
• Small Dynamical Heights for Quadratic Polynomials and Rational Functions

  2013cites this paper
• Misiurewicz points for polynomial maps and transversality

  2013cites this paper