We address 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 $O_{X, \varepsilon} (B^{\dim X+\varepsilon})$ for the number of rational points of height at most $B$ on any integral subvariety $X$ of ${\mathbb P}^n_{\mathbb Q}$ of degree $d\geq 2$, where one can write $O_{d,n, \varepsilon}$ instead of $O_{X, \varepsilon}$ as soon as $d\geq 4$. Our main contribution is to remove the factor $B^\varepsilon$ as soon as $d \geq 5$, without introducing a factor $\log B$, while moreover obtaining polynomial dependence on $d$ of the implied constant. Working polynomially in $d$ allows us to give a self-contained and slightly simplified treatment of dimension growth for degree $d \geq 16$, while in the range $5 \leq d \leq 15$ we invoke results by Browning, Heath-Brown and Salberger. 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. The former improvement 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 which brings in many primes in Heath-Brown's variant of the determinant method, and on recent work by Walsh and Ellenberg--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 $d$.

中文翻译：

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维数增长猜想，多项式的次数和无对数因子

我们解决了 Heath-Brown 和 Serre 的维度增长猜想（由 Salberger 证明），当度数 $d$ 增长时。回想一下，Salberger 的维度增长结果给出了形式 $O_{X, \varepsilon} (B^{\dim X+\varepsilon})$ 的边界，用于任何整数子变量 $X 上的至多 $B$ 的有理高度点数${\mathbb P}^n_{\mathbb Q}$ 的度数 $d\geq 2$，其中可以写成 $O_{d,n, \varepsilon}$ 而不是 $O_{X, \varepsilon} $ 尽快 $d\geq 4$。我们的主要贡献是在 $d \geq 5$ 时立即移除因子 $B^\varepsilon$，而不引入因子 $\log B$，同时获得隐含常数 $d$ 的多项式依赖。在 $d$ 中进行多项式工作允许我们对度数 $d \geq 16$ 的维度增长进行独立且略微简化的处理，而在 $5 \leq d \leq 15$ 范围内，我们调用了 Browning、Heath-Brown 和 Salberger 的结果。在此过程中，我们改进了 Bombieri 和 Pila 关于仿射曲线上有界高度的积分点数的众所周知的界限，以及 Walsh 关于射影曲线上有界高度的有理点数的界限。由于 Bhargava、Shankar、Taniguchi、Thorne、Tsimerman 和 Zhao 对 $d$ 数字字段的类组的 $2$-torsion 子组的大小，前一项改进导致最近的估计略有提高。我们的处理建立在 Salberger 最近的工作之上，该工作引入了 Heath-Brown 行列式方法的变体中的许多素数，以及 Walsh 和 Ellenberg--Venkatesh 的近期工作，他们引入了定义多项式的大小。