2025/10/18 by Lea Beneish, Andrew Granville, Beneish, Lea +1
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2510.16649
openalex publication_date 2025/10/18 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28
We study the set of D such that a given irreducible hypersurface C of degree d has infinitely many points of degree D over 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 ≫ x1/2d2-ε distinct fields with degree D and discriminant ≤ 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.