My research interests are generally in the areas of Arithmetic Geometry, Algebraic Number Theory, and Arithmetic Dynamics. I'm interested in both theoretical and computational aspects of these subjects; in fact, most of my work is a blend of theory and computation. At present, I'm mainly working on various Galois-theoretic problems in dynamics. This survey article is a good reference to learn more about the field of Arithmetic Dynamics and some of the main open problems in the area.

Publications

14. Portraits of quadratic rational maps with a small critical cycle.(arXiv)

With Tyler Dunaisky. J. Number Theory 275 (2025), 135-159.

13. Algebraic periodic points of transcendental entire functions.(arXiv)

With Diego Marques, Carlos Gustavo Moreira, and Pavel Trojovský.
Int. J. Number Theory. To appear.

12. Dynatomic Galois groups for a family of quadratic rational maps.(arXiv)

With Allan Lacy. Int. J. Number Theory 20 (2024), no. 7, 1701-1724.

11. Quadratic points on dynamical modular curves.(arXiv)

With John Doyle. Res. number theory 10,59 (2024).

10. Morikawa's unsolved problem.(PDF)

With Jan Holly. Amer. Math. Monthly 128 (2021), no. 3, 214-237.

9. Twists of hyperelliptic curves by integers in progressions modulo p.(PDF)

With Paul Pollack. Acta Arith. 192 (2020), no. 1, 63-71.

8. A finiteness theorem for specializations of dynatomic polynomials.(arXiv)

Algebra Number Theory 13 (2019), no. 4, 963-993.

7. Galois groups over rational function fields and explicit Hilbert irreducibility.(arXiv)

With Nicole Sutherland. J. Symbolic Comput. 103 (2021), 108-126.
An implementation of the main algorithm is included in Magma via the intrinsic function HilbertIrreducibilityCurves.

6. Galois groups in a family of dynatomic polynomials.(arXiv)

J. Number Theory 187 (2018), 469-511.

5. A local-global principle in the dynamics of quadratic polynomials.(arXiv)

Int. J. Number Theory 12 (2016), no. 8, 2265-2297.

4. Squarefree parts of polynomial values.(PDF)

J. Théor. Nombres Bordeaux 28 (2016), 699-724.

3. Computing points of bounded height in projective space over a number field.(arXiv)

Math. Comp. 85 (2016), 423-447.

2. Computing algebraic numbers of bounded height.(arXiv)

With John Doyle. Math. Comp. 84 (2015), 2867-2891.
An implementation of the main algorithm is included in Sage via the number field function bdd_height.

1. Preperiodic points for quadratic polynomials over quadratic fields.(PDF)

With John Doyle and Xander Faber. New York J. Math. 20 (2014), 507-605.

Doctoral dissertation

Quadratic points on modular curves (PDF). Advisor: Dino Lorenzini