Determination of all rational preperiodic points for morphisms of PN

TL;DR

This paper introduces an algorithm to find all rational preperiodic points of morphisms on projective space over number fields, implemented in Sage, and generalizes dynatomic cycles to preperiodic points.

Contribution

It provides a new effective algorithm for determining rational preperiodic points and extends the concept of dynatomic cycles to include preperiodic points.

Findings

Algorithm successfully determines all rational preperiodic points.

Implementation available in Sage software.

Generalization of dynatomic cycles to preperiodic points is effective.

Abstract

For a morphism $f:\P^N \to \P^N$, 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.