多项式向量场轨道的尖锐零点估计
Sharp zero estimates for trajectories of polynomial vector fields
- Weizmann Institute of Science（魏茨曼科学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对多项式向量场的轨迹，在全局D-性质下，用数域和对数高度推广了Nesterenko的零点估计，证明了紧致集中零点数目的本质上最优界。
AI中文摘要：
若 $f$ 是满足一个代数常微分方程的函数元组，且 $P\in{\mathbb C}(x)[f]$，在超越数论的应用中，通常考虑 $P(x,f)$ 在给定点处的零点阶数的上界，该上界以 $\operatorname{deg}_x P$ 和 $\operatorname{deg}_f P$ 表示。Nesterenko 引入了称为 D-性质的条件，该条件在许多应用中成立，并在此条件下证明了本质上最优的界。我们证明了该结果的全局形式，其中域 $\mathbb C(x)$ 被数域 $K$ 替换，$\operatorname{deg}_x P$ 被对数高度 $\operatorname{h}(P)$ 替换。在类似的全局 D-性质下，我们证明了在固定紧致集中按重数计数的零点数目的本质上最优的界。该结果在点计数定理中的应用在与 Hirata-Kohno 和 Kawashima 合作的另一篇论文中展开。
英文摘要：
If $f$ is a tuple of functions satisfying an algebraic ODE and $P\in{\mathbb C}(x)[f]$, it is common in applications to transcendental number theory to consider upper bounds for the order of zero of $P(x,f)$ at a given point in terms of $\operatorname{deg}_x P,\operatorname{deg}_f P$. Nesterenko introduced a condition known as the D-property, which holds in many applications, and proved essentially optimal bounds under this condition. We prove a global form of this result, where the field $\mathbb C(x)$ is replaced by a number field $K$ and $\operatorname{deg}_x P$ is replaced by the logarithmic height $\operatorname{h}(P)$. Under an analogous global D-property, we prove essentially optimal bounds for the number of zeroes, counted with multiplicities, in a fixed compact set. Applications of this result to point-counting theorems are developed in a separate joint paper with Hirata-Kohno and Kawashima.
