Degrees of points on irreducible hypersurfaces

Abstract

We study the set of $D$ such that a given irreducible hypersurface $C$ of degree $d$ has infinitely many points of degree $D$ over $\mathbb{Q}$. We give a new explicit proof that this set contains all (positive) multiples of the index of $C$ with finitely many exceptions. When $D$ is sufficiently large and divisible by the index of $C$, we show there are $\gg x^{1/2d^2-ε}$ distinct fields with degree $D$ and discriminant $\leq x$ containing new non-singular points on $C$. Our proof relies on (what we define to be) the index of the Newton polytope $H(C)$ for $C$ which we use as combinatorial proxy for the index of $C$. We conjecture that for almost all $C$ with a given Newton polytope $H$, the index of $H$ equals the index of $C$ and we prove this conjecture for a positive proportion of curves with $H(C)=H$. As an application of our techniques, we prove half of Bhargava's conjecture on the least odd degree of points on a typical hyperelliptic and we recover Springer's theorem and a related statement for rational points on cubic hypersurfaces.