J. Doyle

activity

20112021

most citedPreperiodic points for quadratic polynomials over quadratic fields

12 citations · 38 across the 14 of their papers we have counts for

collaborators

19 papers

math.AG2021

Dynamical moduli spaces and polynomial endomorphisms of configurations

Talia Blum, John R. Doyle, Trevor Hyde +3

A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes…

math.NT2021

Dynatomic polynomials, necklace operators, and universal relations for dynamical units

John R. Doyle, Paul Fili, Trevor Hyde

Given a generic polynomial $f (x)$ , the generalized dynatomic polynomial $Φ_{f, c, d} (x)$ vanishes at precisely those $α$ such that $f^{c} (α)$ has period exactly $d$ under iteration of…

math.NT2021

Galois groups and prime divisors in random quadratic sequences

John R. Doyle, Vivian Olsiewski Healey, Wade Hindes +1

Given a set $S = {x^{2} + c_{1}, \dots, x^{2} + c_{s}}$ defined over a field and an infinite sequence $γ$ of elements of $S$ , one can associate an arboreal representation to $γ$ , generalizing th…

math.NT2020

Multivariate Polynomial Values in Difference Sets

John R. Doyle, Alex Rice

For $ℓ \geq 2$ and $h \in Z [x_{1}, \dots, x_{ℓ}]$ of degree $k \geq 2$ , we show that every set $A \subseteq {1, 2, \dots, N}$ lacking nonzero differences in $h(\mathbb{Z}^{\e…

math.NT2019★ 5 cited

A question for iterated Galois groups in arithmetic dynamics

Andrew Bridy, John R. Doyle, Dragos Ghioca +2

We formulate a general question regarding the size of the iterated Galois groups associated to an algebraic dynamical system and then we discuss some special cases of our question.

math.NT2018★ 4 cited

Moduli Spaces for Dynamical Systems with Portraits

John R. Doyle, Joseph H. Silverman

A $portrait$ $P$ on $P^{N}$ is a pair of finite point sets $Y \subseteq X \subset P^{N}$ , a map $Y \to X$ , and an assignment of weights to the points…

math.NT2018

Finite index theorems for iterated Galois groups of unicritical polynomials

Andrew Bridy, John R. Doyle, Dragos Ghioca +2

Let $K$ be the function field of a smooth, irreducible curve defined over $Q$ . Let $f \in K [x]$ be of the form $f (x) = x^{q} + c$ where $q = p^{r}, r \geq 1,$ is a powe…

math.NT2018★ 1 cited

A Uniform Field-of-Definition/Field-of-Moduli Bound for Dynamical Systems on $P^{N}$

John R. Doyle, Joseph H. Silverman

Let $f : P^{N} \to P^{N}$ be an endomorphism of degree $d \geq 2$ defined over $Q$ or $Q_{p}$ , and let $K$ be the field of moduli of $f…

math.DS2018

Preperiodic points for quadratic polynomials over cyclotomic quadratic fields

John R. Doyle

Given a number field $K$ and a polynomial $f (z) \in K [z]$ of degree at least 2, one can construct a finite directed graph $G (f, K)$ whose vertices are the $K$ -rational preperiodic p…

math.NT2017★ 10 cited

Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

John R. Doyle, Bjorn Poonen

Fix $d \geq 2$ and a field $k$ such that $char k ∤ d$ . Assume that $k$ contains the $d$ th roots of $1$ . Then the irreducible components of the curves over $k$ parameter…

math.DS2017

Dynamical modular curves for quadratic polynomial maps

John R. Doyle

Motivated by the dynamical uniform boundedness conjecture of Morton and Silverman, specifically in the case of quadratic polynomials, we give a formal construction of a certain cla…

math.DS2017★ 5 cited

Reduction of dynatomic curves

John R. Doyle, Holly Krieger, Andrew Obus +3

The dynatomic modular curves parametrize polynomial maps together with a point of period $n$ . It is known that the dynatomic curves $Y_{1} (n)$ are smooth and irreducible in character…

math.DS2016

Preperiodic portraits for unicritical polynomials over a rational function field

John R. Doyle

Let $K$ be an algebraically closed field of characteristic zero, and let $K := K (t)$ be the rational function field over $K$ . For each $d \geq 2$ , we consider the unicriti…

math.NT2015

Preperiodic points for quadratic polynomials with small cycles over quadratic fields

John R. Doyle

Given a number field $K$ and a polynomial $f (z) \in K [z]$ , one can naturally construct a finite directed graph $G (f, K)$ whose vertices are the $K$ -rational preperiodic points of $f…

math.NT2015

Configuration of the Crucial Set for a Quadratic Rational Map

John R. Doyle, Kenneth Jacobs, Robert Rumely

Let $K$ be a complete, algebraically closed non-archimedean valued field, and let $φ (z) \in K (z)$ have degree two. We describe the crucial set of $φ$ in terms of the multipliers of…

math.DS2015

Preperiodic portraits for unicritical polynomials

John R. Doyle

Let $K$ be an algebraically closed field of characteristic zero, and for $c \in K$ and an integer $d \geq 2$ , define $f_{d, c} (z) := z^{d} + c \in K [z]$ . We consider the following ques…

math.NT2013★ 12 cited

Preperiodic points for quadratic polynomials over quadratic fields

John R. Doyle, Xander Faber, David Krumm

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 p…

math.NT2011★ 1 cited

Computing algebraic numbers of bounded height

John R. Doyle, David Krumm

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

math.MG2011

Apollonian circle packings of the half-plane

Michael Ching, John R. Doyle

We consider Apollonian circle packings of a half Euclidean plane. We give necessary and sufficient conditions for two such packings to be related by a Euclidean similarity (that is…

Dynamical moduli spaces and polynomial endomorphisms of configurations

Dynatomic polynomials, necklace operators, and universal relations for dynamical units

Galois groups and prime divisors in random quadratic sequences

Multivariate Polynomial Values in Difference Sets

A question for iterated Galois groups in arithmetic dynamics

Moduli Spaces for Dynamical Systems with Portraits

Finite index theorems for iterated Galois groups of unicritical polynomials

A Uniform Field-of-Definition/Field-of-Moduli Bound for Dynamical Systems on PN

Preperiodic points for quadratic polynomials over cyclotomic quadratic fields

Gonality of dynatomic curves and strong uniform boundedness of preperiodic points

Dynamical modular curves for quadratic polynomial maps

Reduction of dynatomic curves

Preperiodic portraits for unicritical polynomials over a rational function field

Preperiodic points for quadratic polynomials with small cycles over quadratic fields

Configuration of the Crucial Set for a Quadratic Rational Map

Preperiodic portraits for unicritical polynomials

Preperiodic points for quadratic polynomials over quadratic fields

Computing algebraic numbers of bounded height

Apollonian circle packings of the half-plane

A Uniform Field-of-Definition/Field-of-Moduli Bound for Dynamical Systems on $P^{N}$