The dimension growth conjecture, polynomial in the degree and without logarithmic factors

Abstract

We study Heath-Brown’s and Serre’s dimension growth conjecture (proved by Salberger) when the degree $d$ grows. Recall that Salberger’s dimension growth results give bounds of the form $O_{X,𝜀}\left(B^{dimX+𝜀}\right)$ for the number of rational points of height at most $B$ on any integral subvariety $X$ of ${\mathbb{P}}_{\mathbb{Q}}^n$ of degree $d\ge 2$, where one can write $O_{d,n,𝜀}$ instead of $O_{X,𝜀}$ as soon as $d\ge 4$. We give the following simplified and strengthened forms of these results: we remove the factor $B^{𝜀}$ as soon as $d\ge 5$, we obtain polynomial dependence on $d$ of the implied constant, and we give a simplified, self-contained approach for $d\ge 16$. Along the way, we improve the well-known bounds due to Bombieri and Pila on the number of integral points of bounded height on affine curves and those by Walsh on the number of rational points of bounded height on projective curves. This leads to a slight sharpening of a recent estimate due to Bhargava, Shankar, Taniguchi, Thorne, Tsimerman and Zhao on the size of the $2$-torsion subgroup of the class group of a degree $d$ number field. Our treatment builds on recent work by Salberger, who brings in many primes in Heath-Brown’s variant of the determinant method, and on recent work by Walsh and by Ellenberg and Venkatesh who bring in the size of the defining polynomial. We also obtain lower bounds showing that one cannot do better than polynomial dependence on $d$.