Heights and preperiodic points of polynomials over function fields

By Robert L. Benedetto. 2005

61 citations recorded by the source corpus.

Let K be a function field in one variable over an arbitrary field F. Given a rational function f(z) in K(z) of degree at least two, the associated canonical height on the projective line was defined by Call and Silverman. The preperiodic points of f all have canonical height zero; conversely, if F is a finite field, then every point of canonical height zero is preperiodic. However, if F is an infinite field, then there may be non-preperiodic points of canonical height zero. In this paper, we show that for polynomial f, such points exist only if f is isotrivial. In fact, such K-rational points exist only if f is defined over the constant field of K after a K-rational change of coordinates.

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1f957c5e91135115b5ca8db05eec8af3df043bfc

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In Algebra and Number Theory

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Bifurcations, intersections, and heights (2014, background): “This was proved for polynomials f in [Benedetto 2005].”
Dynamical Degrees, Arithmetic Degrees, and Canonical Heights for Dominant Rational Self-Maps of Projective Space (2011, background): “The situation over function fields is subtler; see for example [7, 13].”
A finiteness theorem for canonical heights attached to rational maps over function fields (2006, background): “This would provide a better value of ε but worse upper bound on N than is given by the argument above.”
Primitive divisors in arithmetic dynamics (2007, background): “Tom Tucker has pointed out to the authors that the results of this paper should be valid also for iteration of non-split functions over C(T ), and more generally over one-dimensional function fields of characteristic 0, since in this setting Benedetto [4] (for polynomial maps) and Baker [1] (for rat”
ABC implies primitive prime divisors in arithmetic dynamics (2012, background): “By works of Benedetto [4] and Baker [1], one has the same result for non-isotrivial rational functions over a function field.”
Preperiodic points and unlikely intersections (2009, background): “[Ben05] Let k be an abstract function field.”