参数化曲线上格点可见性的局部-全局原理
Local-global principles for visibility of lattice points on parameterized curves
- Indraprastha Institute of Information Technology Delhi（德里印度理工学院）
- Chennai Mathematical Institute（金奈数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文针对参数化曲线上的格点可见性，建立局部-全局原理，证明加权齐次族及单项式情形下可见性可逐素数判定，并给出密度乘积公式及非可见点数量的改进界。
中文摘要 AI 辅助
我们发展了参数化曲线族上格点可见性的局部-全局理论。我们引入了$p$-adic可见性的概念，并询问一个格点是否在全局可见，当且仅当它在每个素数处可见。我们证明了这一局部-全局原理对于一大类其参数化关于正权重是齐次的族成立。当这些曲线上的点填满整个正整数格时，我们计算了可见点的局部和全局密度，并表明全局密度是局部密度的乘积。然后我们考虑形如$y=qP(x)$的多项式族，其中$q\in\mathbb{Q}_{>0}$，并表明当且仅当$P$是单项式时，可见性可以逐素数检测。对于非单项式多项式，局部-全局原理可能失效，但失效点的集合密度为零；对于可分多项式，我们还获得了非可见点数量的定量界，改进了先前已知的界。我们进一步考虑其格点位于环境空间的真低维代数子集上的族，并表明它们的可见性密度可能与完整格的可见性密度表现不同。最后，我们将理论从从原点可见性扩展到从一个格点到另一个格点的可见性，并表明相应的局部-全局原理对于加权齐次族仍然成立。
英文摘要
We develop a local-global theory for visibility of lattice points on families of parameterized curves. We introduce a notion of $p$-adic visibility and ask whether a lattice point is globally visible precisely when it is visible at every prime. We prove that this local-global principle holds for a broad class of families whose parametrizations are homogeneous with respect to positive weights. When the points on these curves fill the entire positive integer lattice, we compute the local and global densities of visible points and show that the global density is the product of the local densities. We then consider polynomial families of the form $y=qP(x)$, with $q\in\mathbb{Q}_{>0}$, and show that visibility can be detected prime by prime exactly when $P$ is a monomial. For non-monomial polynomials the local-global principle can fail, but the set of points where it fails has density zero; for separable polynomials we also obtain a quantitative bound for the number of non-visible points, improving the previously known bound. We further consider families whose lattice points lie on a proper lower-dimensional algebraic subset of the ambient space and show that their visibility densities can behave differently from those of the full lattice. Finally, we extend the theory from visibility from the origin to visibility from one lattice point to another and show that the corresponding local-global principle continues to hold for weighted homogeneous families.
