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Let $f in Q(z)$ be a polynomial or rational function of degree 2. A special case of Morton and Silvermans Dynamical Uniform Boundedness Conjecture states that the number of rational preperiodic points of $f$ is bounded above by an absolute constant. A related conjecture of Silverman states that the canonical height $hat{h}_f(x)$ of a non-preperiodic rational point $x$ is bounded below by a uniform multiple of the height of $f$ itself. We provide support for these conjectures by computing the set of preperiodic and small height rational points for a set of degree 2 maps far beyond the range of previous searches.
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For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article ``Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems, by two of the present authors and five others, it was shown that the number of rational iterated pre-images of the origin is bounded as one varies the morphism in a certain one-dimensional family. Subject to the validity of the Birch and Swinnerton-Dyer conjecture and some other related conjectures for the L-series of a specific abelian variety and using a number of modern tools for locating rational points on high genus curves, we show that the maximum number of rational iterated pre-images is six. We also provide further insight into the geometry of the ``pre-image curves.
Let f in Q[z] be a polynomial of degree d at least two. The associated canonical height hat{h}_f is a certain real-valued function on Q that returns zero precisely at preperiodic rational points of f. 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, hat{h}_f is bounded below by a positive constant (depending only on d) times some kind of height of f 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.
Given a generic polynomial $f(x)$, the generalized dynatomic polynomial $Phi_{f,c,d}(x)$ vanishes at precisely those $alpha$ such that $f^c(alpha)$ has period exactly $d$ under iteration of $f(x)$. We show that the shifted dynatomic polynomials $Phi_{f,c,d}(x) - 1$ often have generalized dynatomic factors, and that these factors are in correspondence with certain cyclotomic factors of necklace polynomials. These dynatomic factors of $Phi_{f,c,d}(x) - 1$ have an interpretation in terms of new multiplicative relations between dynamical units which are uniform in the polynomial $f(x)$.
We give necessary and sufficient conditions for post-critically finite polynomials to have potential good reduction at a given prime. We also answer in the negative a question posed by Silverman about conservative polynomials. Both proofs rely on dynamical Belyi polynomials as exemplars of PCF (resp. conservative) maps.
We give algorithms for computing the singular moduli of suitable nonholomorphic modular functions F(z). By combining the theory of isogeny volcanoes with a beautiful observation of Masser concerning the nonholomorphic Eisenstein series E_2*(z), we obtain CRT-based algorithms that compute the class polynomials H_D(F;x), whose roots are the discriminant D singular moduli for F(z). By applying these results to a specific weak Maass form F_p(z), we obtain a CRT-based algorithm for computing partition class polynomials, a sequence of polynomials whose traces give the partition numbers p(n). Under the GRH, the expected running time of this algorithm is O(n^{5/2+o(1)}). Key to these results is a fast CRT-based algorithm for computing the classical modular polynomial Phi_m(X,Y) that we obtain by extending the isogeny volcano approach previously developed for prime values of m.
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