Determination of all rational preperiodic points for morphisms of PN

Hutz, Benjamin

doi:10.1090/S0025-5718-2014-02850-0

Determination of all rational preperiodic points for morphisms of PN
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Math. Comp. 84 (2015), 289-308 Request permission

Abstract:

For a morphism $f:\mathbb {P}^N \to \mathbb {P}^N$, 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 $\mathbb {Q}$. 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.

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