论文标题
曲线上有界高度的点和尺寸增长的猜想在$ \ mathbb {f} _q [t] $上
Points of bounded height on curves and the dimension growth conjecture over $\mathbb{F}_q[t]$
论文作者
论文摘要
在本文中，我们证明了几个新的统一上限，这些统一的上限在$ \ mathbb {f} _q [t] $上的有限高度点的数量上。对于投影曲线，我们证明了沃尔什（Walsh）结果的类似物，其依赖于$ q $和曲线的$ d $。对于仿射曲线，这可以改善Sedunova的边界，以及Cluckers，Forey和Loeser。在较高的维度中，我们证明了$ d \ geq 64 $的高度曲面的维度增长，这是Castryck，Cluckers，Dittmann和Nguyen在特征零中的工作。这些界限在多项式取决于$ q $和$ d $，而正是这种依赖性简化了维度生长猜想的处理。
In this article we prove several new uniform upper bounds on the number of points of bounded height on varieties over $\mathbb{F}_q[t]$. For projective curves, we prove the analogue of Walsh' result with polynomial dependence on $q$ and the degree $d$ of the curve. For affine curves, this yields an improvement to bounds by Sedunova, and Cluckers, Forey and Loeser. In higher dimensions, we prove a version of dimension growth for hypersurfaces of degree $d\geq 64$, building on work by Castryck, Cluckers, Dittmann and Nguyen in characteristic zero. These bounds depend polynomially on $q$ and $d$, and it is this dependence which simplifies the treatment of the dimension growth conjecture.
