AI 中文总结
该数学研究针对$p,q,r\geq2$时的Campana orbifold，建立有界高度本原正Campana点的上界，给出广义费马曲面不对称盒子中本原整点的一致估计，获优于平凡界的幂次节省。
AI 中文摘要
设$p,q,r\geq2$，考虑Campana orbifold $\left( \mathbb{P}^1, \left(1-\tfrac1p\right)[0] +\left(1-\tfrac1q\right)[1] +\left(1-\tfrac1r\right)[\infty] \right)$。该orbifold上的本原正Campana点对应$a+b=c$的解，其中$a,b,c$分别为$p$-full、$q$-full、$r$-full。我们在大范围指数中建立有界高度此类点的上界，其优于平凡界的幂次节省。主要分析输入是广义费马曲面$a_1x^p+a_2y^q+a_3z^r=0$的不对称盒子中本原整点的估计，该估计对系数一致。
英文摘要
Let $p,q,r\geq 2$ and consider the Campana orbifold \[ \left( \mathbb{P}^1, \left(1-\tfrac1p\right)[0] +\left(1-\tfrac1q\right)[1] +\left(1-\tfrac1r\right)[\infty] \right). \] Primitive positive Campana points on this orbifold correspond to solutions of $a+b=c$ in which $a$, $b$, and $c$ are respectively $p$-full, $q$-full, and $r$-full. We establish upper bounds for the number of such points of bounded height in a broad range of exponents, with a power-saving over the trivial bound. The main analytic input is an estimate for primitive integral points in lopsided boxes on generalized Fermat surfaces $a_1x^p+a_2y^q+a_3z^r=0$, which is uniform in the coefficients.
Comments18 pages
