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- Title
Preperiodic points for quadratic polynomials with small cycles over quadratic fields.
- Authors
Doyle, John R.
- Abstract
Given a number field K and a polynomial f(z)∈K[z]<inline-graphic></inline-graphic>, one can naturally construct a finite directed graph G(f, K) whose vertices are the K-rational preperiodic points of f, with an edge α→β<inline-graphic></inline-graphic> if and only if f(α)=β<inline-graphic></inline-graphic>. The dynamical uniform boundedness conjecture of Morton and Silverman suggests that, for fixed integers n≥1<inline-graphic></inline-graphic> and d≥2<inline-graphic></inline-graphic>, there are only finitely many isomorphism classes of directed graphs G(f, K) as one ranges over all number fields K of degree n and polynomials f(z)∈K[z]<inline-graphic></inline-graphic> of degree d. In the case (n,d)=(1,2)<inline-graphic></inline-graphic>, Poonen has given a complete classification of all directed graphs which may be realized as G(f,Q)<inline-graphic></inline-graphic> for some quadratic polynomial f(z)∈Q[z]<inline-graphic></inline-graphic>, under the assumption that f does not admit rational points of large period. The purpose of the present article is to continue the work begun by the author, Faber, and Krumm on the case (n,d)=(2,2)<inline-graphic></inline-graphic>. By combining the results of the previous article with a number of new results, we arrive at a partial result toward a theorem like Poonen’s—with a similar assumption on points of large period—but over all quadratic extensions of Q<inline-graphic></inline-graphic>.
- Publication
Mathematische Zeitschrift, 2018, Vol 289, Issue 1/2, p729
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-017-1973-1