arXiv subjects

Alexander Galarraga

Publications and source records attributed to Alexander Galarraga.

5 recordsLinked to original sources

arXiv2026

Indecomposable translates and degree-$d$ points

A curve over a number field has infinitely many degree-$d$ points if and only if there exists a degree-$d$ $\mathbb{P}^1$- or $AV$-parametrized point. We show that a $\mathbb{P}^1$-isolated $AV$-parametrized point arises only from an abelian translate satisfying a certain property, which we term ``indecomposable". We apply our results to a family of genus-7 curves for which we can show that every effective abelian translate in degree 5 is decomposable over the algebraic closure, giving a negative answer to a question of Viray and Vogt.

math.NT↗

arXiv2025

Superelliptic degree sets over Henselian fields

Let $K$ be a discretely valued Henselian field. Creutz and Viray show that the degree set of a curve $C$ over a $p$-adic field can miss infinitely many multiples of the index of $C$, a phenomenon that cannot occur over finitely generated fields. For curves $C/K$ with a cyclic cover of $\mathbb{P}^1$ of prime degree, under mild assumptions, we completely characterize how and when this behavior can occur, and give a method for computing degree sets of curves of this type.

math.NT↗

arXiv2025

Cubic points on dynamical modular curves

We consider the family of dynamical modular curves associated to quadratic polynomial maps and determine precisely which of these curves have infinitely many cubic points. We use this to prove a classification statement on preperiodic points for quadratic polynomials over cubic fields, extending previous work of Poonen, Faber, and the first author and Krumm.

math.NT↗

arXiv2022

Integrality and Thurston Rigidity for Bicritical PCF Polynomials

We give an algebraic proof of an important consequence of Thurston rigidity for bicritical PCF polynomials with periodic critical points under certain mild assumptions. The key result is that when the family of bicritical polynomials is parametrized using dynamical Belyi polynomials, the PCF solutions are integral at certain special primes, which we term ``index divisor free primes.'' We prove the existence of index divisor free primes in all but finitely many cases and conjecture the complete list of exceptions. These primes are then used to prove transversality.

math.DS↗

arXiv2022

New Normal Forms For Degree Three Polynomials and Rational Functions

When studying families in the moduli space of dynamical systems, choosing an appropriate representative function for a conjugacy class can be a delicate task. The most delicate questions surround rationality of the conjugacy class compared to rationality of the defining polynomials of the representation. We give a normal form for degree three polynomials which has the property that the set of fixed points is equal to the set of fixed point multipliers. This normal form is given in terms of moduli space invariants and, hence, has nice rationality properties. We further classify all degree three rational maps which can be conjugated to have a similar relationship between the fixed points and the fixed point multipliers.

math.DS↗