Determination of all rational preperiodic points for morphisms of PN

Abstract

For a morphism f : P N → P N f:\mathbb {P}^N \to \mathbb {P}^N , the points whose forward orbit by f f is finite are called preperiodic points for f f . This article presents an algorithm to effectively determine all the rational preperiodic points for f f defined over a given number field K K . This algorithm is implemented in the open-source software Sage for Q \mathbb {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.