John R. Doyle

Computing algebraic numbers of bounded height

We describe an algorithm for listing all elements of bounded height in a given number field.

Reduction of dynatomic curves

The dynatomic modular curves parametrize polynomial maps together with a point of period

Preperiodic portraits for unicritical polynomials

n

Preperiodic points for quadratic polynomials over quadratic fields

. It is known that the dynatomic curves

Configuration of the crucial set for a quadratic rational map

Y_1(n)

Preperiodic portraits for unicritical polynomials over a rational function field

are smooth and irreducible in characteristic 0 for families of polynomial maps of the form

Dynamical modular curves for quadratic polynomial maps

f_c(z) = z^m +c

Preperiodic points for quadratic polynomials with small cycles over quadratic fields

where

Apollonian circle packings of the half-plane

m\geq 2

Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

. In the present paper, we build on the work of Morton to partially characterize the primes