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Degrees of points on irreducible hypersurfaces

Lea Beneish, Andrew Granville

TL;DR

The paper develops a quantitative Diophantine framework connecting the degrees of non-singular points on irreducible hypersurfaces to the geometric shape of the defining polynomial via the Newton polytope. It introduces the index $G(C)$ and a combinatorial proxy $H(C)$, defines Exp$(C)$ through RIH and Frobenius-closure arguments, and proves explicit lower bounds for the number of fields of degree $D$ containing new points on $C$, with $D$-degrees growing as $X^{\frac{1}{2d^2}-\varepsilon}$ in many cases. A central contribution is the explicit mechanism to construct many distinct fields from polynomial substitutions, alongside an effective description of the exceptional set $\mathcal{E}(C)$ and its relation to $G(C)$ and $H(C)$; this yields new insights into the sparsity of odd-degree points and recovers classical results like Springer’s theorem while supporting conjectures of Bhargava and collaborators in specific families, notably hyperelliptic curves. The framework unifies combinatorial, algebraic, and analytic methods to quantify how the “shape” of a hypersurface governs arithmetic point distribution, with wide-ranging applications to hyperelliptic curves and low-degree diagonal and superelliptic hypersurfaces. The results provide a pathway to further explicit estimates and conjectures about degree distribution of algebraic points on varieties in Diophantine geometry.

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.

Degrees of points on irreducible hypersurfaces

TL;DR

The paper develops a quantitative Diophantine framework connecting the degrees of non-singular points on irreducible hypersurfaces to the geometric shape of the defining polynomial via the Newton polytope. It introduces the index and a combinatorial proxy , defines Exp through RIH and Frobenius-closure arguments, and proves explicit lower bounds for the number of fields of degree containing new points on , with -degrees growing as in many cases. A central contribution is the explicit mechanism to construct many distinct fields from polynomial substitutions, alongside an effective description of the exceptional set and its relation to and ; this yields new insights into the sparsity of odd-degree points and recovers classical results like Springer’s theorem while supporting conjectures of Bhargava and collaborators in specific families, notably hyperelliptic curves. The framework unifies combinatorial, algebraic, and analytic methods to quantify how the “shape” of a hypersurface governs arithmetic point distribution, with wide-ranging applications to hyperelliptic curves and low-degree diagonal and superelliptic hypersurfaces. The results provide a pathway to further explicit estimates and conjectures about degree distribution of algebraic points on varieties in Diophantine geometry.

Abstract

We study the set of such that a given irreducible hypersurface of degree has infinitely many points of degree over . We give a new explicit proof that this set contains all (positive) multiples of the index of with finitely many exceptions. When is sufficiently large and divisible by the index of , we show there are distinct fields with degree and discriminant containing new non-singular points on . Our proof relies on (what we define to be) the index of the Newton polytope for which we use as combinatorial proxy for the index of . We conjecture that for almost all with a given Newton polytope , the index of equals the index of and we prove this conjecture for a positive proportion of curves with . 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.
Paper Structure (34 sections, 40 theorems, 100 equations)

This paper contains 34 sections, 40 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. General quantitative results
  3. A combinatorial proxy for the index
  4. A conjecture: More than a proxy
  5. Applications to the sparsity of odd degree points
  6. Applications to hypersurfaces
  7. Applications to hyperelliptic curves
  8. The reasonable image hypothesis and Exp($C$)
  9. The reasonable image hypothesis
  10. Sub-images
  11. Working with corners
  12. A more geometric perspective
  13. Elementary number theory
  14. Elements of $D(H)^{\text{inf}}$ constructed from a given element of $H_f$
  15. Defining $D(H)^{\text{inf}}$
  16. ...and 19 more sections

Key Result

Theorem 1.1

There exists a finite set of integers $\mathcal{E}(C)$ such that

Theorems & Definitions (79)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9: Springer
  • Proposition 1.10: Coray
  • ...and 69 more