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**Preperiodic points for quadratic polynomials over quadratic fields.**
*(English)*
Zbl 1391.37082

Summary: To each quadratic number field \(K\) and each quadratic polynomial \(f\) with \(K\)-coefficients, one can associate a finite directed graph \(G(f, K)\) whose vertices are the \(K\)-rational preperiodic points for \(f\), and whose edges reflect the action of \(f\) on these points. This paper has two main goals. (1) For an abstract directed graph \(G\), classify the pairs \((K, f )\) such that the isomorphism class of \(G\) is realized by \(G(f, K)\). We succeed completely for many graphs \(G\) by applying a variety of dynamical and Diophantine techniques. (2) Give a complete description of the set of isomorphism classes of graphs that can be realized by some \(G(f, K)\). A conjecture of Morton and Silverman implies that this set is finite. Based on our theoretical considerations and a wealth of empirical evidence derived from an algorithm that is developed in this paper, we speculate on a complete list of isomorphism classes of graphs that arise from quadratic polynomials over quadratic fields.

### MSC:

37P35 | Arithmetic properties of periodic points |

14G05 | Rational points |

37P05 | Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps |