arXiv subjects

Joseph Gunther

Publications and source records attributed to Joseph Gunther.

4 recordsLinked to original sources

arXiv2019

Irrational points on random hyperelliptic curves

We consider genus $g$ hyperelliptic curves over $\mathbb{Q}$ with a rational Weierstrass point, ordered by height. If $d < g$ is odd, we prove, under an assumption, that there exists $B_d$ such that a positive proportion of these curves have at most $B_d$ points of degree $d$. If $d < g$ is even, we conditionally bound degree $d$ points not pulled back from points of degree $d/2$ on the projective line. Furthermore, we show one may take $B_2=24$ and B_3=114$. Our proofs proceed by refining recent work of Park, which applied tropical geometry to symmetric power Chabauty, and then applying results of Bhargava and Gross on average ranks of Jacobians of hyperelliptic curves.

math.NT↗

arXiv2016

Slicing the stars: counting algebraic numbers, integers, and units by degree and height

Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree $d$ and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over $\mathbb{Z}$) in a homogeneously expanding star body in $\mathbb{R}^{d+1}$. The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one "slice" corresponding to monic polynomials -- this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which allows us to count units, algebraic integers of given norm, trace, norm and trace, and more. We also refine the lattice point-counting arguments of Chern-Vaaler to obtain explicit error terms with better power savings, which lead to explicit versions of some results of Masser-Vaaler and Barroero.

math.NT↗

arXiv2016

Integral points of bounded degree on the projective line and in dynamical orbits

Let $D$ be a non-empty effective divisor on $\mathbb{P}^1$. We show that when ordered by height, any set of $(D,S)$-integral points on $\mathbb{P}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $φ(z)\in \overline{\mathbb{Q}}(z)$ be a rational function of degree at least two whose second iterate $φ^2(z)$ is not a polynomial. We show that as we vary over points $P\in\mathbb{P}^1(\overline{\mathbb{Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $P$ is absolutely bounded and zero on average.

math.NT↗

arXiv2016

Random hypersurfaces and embedding curves in surfaces over finite fields

We use Poonen's closed point sieve to prove two independent results. First, we show that the obvious obstruction to embedding a curve in a smooth surface is the only obstruction over a perfect field, by proving the finite field analogue of a Bertini-type result of Altman and Kleiman. Second, we prove a conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities.

math.NT↗