Research

My primary field of research is Arithmetic Dynamics, which is the study of number theoretic properties of dynamical systems. In particular, I study number fields known as iterated extensions, which are named for their construction. Namely, if f is a monic polynomial with integer coefficients, and fn denotes the n-fold composition of f with itself, the iterated extensions are the number fields generated by the roots of fn. The roots of fn+1 are algebraic over the splitting field of fn, and thus the field defined by fn+1 contains that defined by fn. In other words, these splitting fields form an infinite tower. I have studied the disciminants of such fields and have found towers in which all the fields are monogenic.

Recently I have been looking at special points on hyperelliptic curves. A point P on an algebraic curve C is a Weierstrass point if the space of functions defined over C with poles restricted to P is larger than what would be expected by the Riemann-Roch theorem. The differences in the dimensions of these spaces (counted by the maximum order of the pole at P) are realizable as a numerical semigroup. Thus by understanding the semigroup, we may classify points on the curve.

I also have an interest in unit groups, and in particular, elliptic units. Following work of Greene and Hajir, I have written a program in PARI/GP which computes an "ideal" generating set of elliptic units in unramified extensions of imaginary quadratic fields. The code is available here: [code]. (There are a few known bugs, so contact me before using.)

Publications

Preprints for most of these articles are available on the arXiv.
  1. with Michael Urbanski, Index divisibility in the orbit of 0 for integral polynomials.
  2. with Hanson Smith and Katherine Stange, A family of monogenic S4 quartic fields arising from elliptic curves.
  3. with Caleb Shor, Characterization of numerical semigroup complements via Apéry sets.
  4. [link] with Annie Chen and Katherine Stange, Index divisibility in dynamical sequences and cyclic orbits modulo p. New York J. Math., 23:1045--1063, 2017.
  5. [link] with Caleb Shor, Weierstrass weights and sums from numerical semigroups. J. Number Theory, 180:45--72, 2017.
  6. [link] A note on the monogeneity of power maps. Albanian J. Math, 11(1):3--12, 2017.
  7. [link] Discriminants of simplest 3n-tic extensions. Funct. Approx. Comment. Math., 53(2):193--214, 2015.
  8. [link] Discriminants of Chebyshev radical extensions. J. Théor Nombres Bordeaux, 26(3):607--633, 2014.
  9. [link] Chebyshev action on finite fields. Disc. Math., 315:83--94, 2014.
  10. [link] with Aaron Yeager, Characterization of the vertex-reinforced random walk and trapping subgraphs. The Pentagon: A Mathematics Magazine for Students, 68(1):21--28, 2008.