David Krumm
YOU?
Author Swipe
View article: Portraits of quadratic rational maps with a small critical cycle
Portraits of quadratic rational maps with a small critical cycle Open
Motivated by a uniform boundedness conjecture of Morton and Silverman, we study the graphs of pre-periodic points for maps in three families of dynamical systems, namely the collections of rational functions of degree two having a periodic…
View article: Algebraic periodic points of transcendental entire functions
Algebraic periodic points of transcendental entire functions Open
We prove the existence of transcendental entire functions $f$ having a property studied by Mahler, namely that $f(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$ and $f^{-1}(\overline{\mathbb{Q}})\subseteq \overline{\mathbb{Q}}$, an…
View article: Dynatomic Galois groups for a family of quadratic rational maps
Dynatomic Galois groups for a family of quadratic rational maps Open
For every nonconstant rational function $ϕ\in\mathbb{Q}(x)$, the Galois groups of the dynatomic polynomials of $ϕ$ encode various properties of $ϕ$ that are of interest in the subject of arithmetic dynamics. We study here the structure of …
View article: Quadratic points on dynamical modular curves
Quadratic points on dynamical modular curves Open
Among all the dynamical modular curves associated to quadratic polynomial maps, we determine which curves have infinitely many quadratic points. This yields a classification statement on preperiodic points for quadratic polynomials over qu…
View article: Morikawa’s Unsolved Problem
Morikawa’s Unsolved Problem Open
By combining theoretical and computational techniques from geometry, calculus, group theory, and Galois theory, we prove the nonexistence of a closed-form algebraic solution to a Japanese geometry problem first stated in the early nineteen…
View article: Twists of hyperelliptic curves by integers in progressions modulo $p$
Twists of hyperelliptic curves by integers in progressions modulo $p$ Open
Let $f(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that the curve $dy^2=f(x)$ has a nontrivial rational or integral …
View article: Galois groups over rational function fields and explicit Hilbert irreducibility
Galois groups over rational function fields and explicit Hilbert irreducibility Open
Let $P\in\mathbb Q[t,x]$ be a polynomial in two variables with rational coefficients, and let $G$ be the Galois group of $P$ over the field $\mathbb Q(t)$. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $c$…
View article: Explicit Hilbert Irreducibility
Explicit Hilbert Irreducibility Open
Let $P(T,X)$ be an irreducible polynomial in two variables with rational coefficients. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $t$ the specialized polynomial $P(t,X)$ is irreducible and has the same …
View article: Computing points of bounded height in projective space over a number field
Computing points of bounded height in projective space over a number field Open
We construct an algorithm for solving the following problem: given a number field $K$, a positive integer $N$, and a positive real number $B$, determine all points in $\mathbb {P}^N(K)$ having relative height at most $B$. A theoretical ana…