Most Odd-Degree Binary Forms Fail to Primitively Represent a Square

TL;DR

This paper demonstrates that for large odd degrees, most superelliptic equations with fixed non-square leading coefficients are insoluble or fail the local-global principle, extending Faltings' finiteness results to a statistical setting.

Contribution

It provides a quantitative asymptotic analysis of the solubility and local-global failure of superelliptic equations of large odd degree, generalizing Faltings' theorem.

Findings

Over 74.9% of equations are insoluble.

Over 71.8% are locally soluble but violate the Hasse principle.

Proportions increase to over 99.9% and 96.7% with more prime divisors of the leading coefficient.

Abstract

Let $F$ be a separable integral binary form of odd degree $N \geq 5$. A result of Darmon and Granville known as ``Faltings plus epsilon'' implies that the degree-$N$ \emph{superelliptic equation} $y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family $\mathscr{F}_N(f_0)$ of degree-$N$ superelliptic equations with fixed leading coefficient $f_0 \in \mathbb{Z} \smallsetminus \pm\mathbb{Z}^2$, ordered by height. For every sufficiently large $N$, we prove that among equations in the family $\mathscr{F}_N(f_0)$, more than $74.9\%$ are insoluble, and more than $71.8\%$ are everywhere locally soluble but fail the Hasse principle due to the Brauer--Manin obstruction. We further show that these proportions rise to at least $99.9\%$ and $96.7\%$, respectively, when $f_0$ has sufficiently many prime divisors of odd multiplicity. Our result can be viewed as a strong asymptotic form of ``Faltings plus epsilon'' for superelliptic equations and constitutes an analogue of Bhargava's result that most hyperelliptic curves over $\mathbb{Q}$ have no rational points.