Arithmetic Dynamics - Number Theoretic Properties of Preperiodic Points

Number Theoretic Properties of Preperiodic Points

Let F(x) be a rational function of degree at least two with coefficients in Q. A theorem of Northcott says that F has only finitely many Q-rational preperiodic points, i.e., F has only finitely many preperiodic points in P1(Q). The Uniform Boundedness Conjecture of Morton and Silverman says that the number of preperiodic points of F in P1(Q) is bounded by a constant that depends only on the degree of F.

More generally, let F : PN → PN be a morphism of degree at least two defined over a number field K. Northcott's theorem says that F has only finitely many preperiodic points in PN(K), and the general Uniform Boundedness Conjecture says that the number of preperiodic points in PN(K) may be bounded solely in terms of N, the degree of F, and the degree of K over Q.

The Uniform Boundedness Conjecture is not known even for quadratic polynomials Fc(x) = x2+c over the rational numbers Q. It is known in this case that Fc(x) cannot have periodic points of period four, five, or six, although the result for period six is contingent on the validity of the conjecture of Birch and Swinnerton-Dyer. Poonen has conjectured that Fc(x) cannot have rational periodic points of any period strictly larger than three.

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