Let φ ∈ Q[z] be a polynomial of degree d at least two. The associated canonical heightĥ φ is a certain real-valued function on Q that returns zero precisely at preperiodic rational points of φ. Morton and Silverman conjectured in 1994 that the number of such points is bounded above by a constant depending only on d. A related conjecture claims that at non-preperiodic rational points,ĥ φ is bounded below by a positive constant (depending only on d) times some kind of height of φ itself. In this paper, we provide support for these conjectures in the case d = 3 by computing the set of small height points for several billion cubic polynomials.In that case, we say x is n-periodic; the smallest such positive integer n is called the period of x. More generally, x is preperiodic under φ if there are integers n > m ≥ 0 such that φ n (x) = φ m (x); equivalently, φ m (x) is periodic for some m ≥ 0.In 1950, using the theory of arithmetic heights, Northcott proved that if deg φ ≥ 2, then φ has only finitely many preperiodic points in Q. (In fact, his result applied far more generally, to morphisms of N-dimensional projective space over any number field.) In 1994, motivated by Northcott's result and by analogies to torsion points of elliptic curves (for which uniform bounds were proven by Mazur [13] over Q and by Merel [14] over arbitrary number fields), Morton and Silverman proposed a dynamical Uniform Boundedness Conjecture [17,18]. Their conjecture applied to the same general setting as Northcott's Theorem, but we state it here only for polynomials over Q.Conjecture 1 (Morton, Silverman 1994). For any d ≥ 2, there is a constant M = M(d) such that no polynomial φ ∈ Q[z] of degree d has more than M rational preperiodic points.Thus far only partial results towards Conjecture 1 have been proven. Several authors [17,18,19,21,24] have bounded the period of a rational periodic point in terms of the smallest prime of good reduction (see Definition 1.3). Others [5,12,15,16,22] have proven that polynomials of degree two cannot have rational periodic points of certain periods by studying the set of rational points on an associated dynamical modular curve; see also [23, Section 4.2]. A different method, introduced in [3] and generalized and sharpened in [2], gave (still non-uniform) bounds for the number of preperiodic points by taking into account all primes, including those of bad reduction.
(Communicated by Bjorn Poonen)For a quadratic polynomial with rational coefficients, we consider the problem of bounding the number of rational points that eventually land at a given constant after iteration, called preimages of the constant. It was shown by Faber, Hutz, Ingram, Jones, Manes, Tucker, and Zieve (2009) that the number of rational preimages is bounded as one varies the polynomial. Explicit bounds on the number of preimages of zero and −1 were addressed in subsequent articles. This article addresses explicit bounds on the number of preimages of any algebraic number for quadratic dynamical systems and provides insight into the geometric surfaces parameterizing such preimages.
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