New York Journal of Mathematics
Volume 25 (2019), 451-466


Kenneth Jacobs and Phillip Williams

Iteration and the minimal resultant

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Published: June 20, 2019.
Keywords: Arithmetic dynamics, Berkovich space, non-archimedean dynamics, minimal resultant, semi-stability.
Subject: Primary 11S82, 37P05; Secondary 37P50.

Let K be an algebraically closed field that is complete with respect to a non-Archimedean absolute value, and let φ ∈ K(z) have degree d ≥ 2. We characterize maps for which the minimal resultant of an iterate φn is given by a simple formula in terms of d, n, and the minimal resultant of φ. Three characterizations of such maps are given, one measure-theoretic and two in terms of the indeterminacy locus I(d) in the parameter space P2d+1 of (possibly degenerate) rational maps. As an application, we are able to give a new explicit formula involving the Arakelov-Green's function attached to φ. We end by illustrating our results with some explicit examples.


The authors would like to thank Laura DeMarco and Matt Baker for helpful coorespondence in preparing this manuscript, along with the anonymous referees for feedback on earlier drafts.

Author information

Kenneth Jacobs:
Northwestern University
2033 Sheridan Rd
Evanston, IL 60202, USA


Phillip Williams:
The King's College
56 Broadway
New York, NY 10004, USA