沿多项式的格点可见性密度为一
Density one for lattice point visibility along polynomials with at least two distinct roots
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中文总结 AI 辅助
本文证明每个至少有两个不同复根的非零整数多项式沿有理斜率曲线的格点可见性密度为一，通过最大公约数截断将问题化为有限个方程并证明解集密度为零，解决了多项式可见性猜想。
中文摘要 AI 辅助
我们证明，每个具有至少两个不同复根的非零整数多项式都具有格点可见性密度一，从而在Lobsenz-Phillips提出的 generality 下解决了Chaubey-Pandey的多项式可见性猜想。可见性沿曲线 y = tF(x) 定义，其中 t 为有理数，若同一条曲线上没有具有更小横坐标的正格点，则该点可见。证明使用最大公约数截断将问题归约为有限多个形如 F(b) = qF(a)（其中 0 < q < 1）的方程；对于每个方程，使得存在正整数解 b < a 的正整数 a 构成的集合密度为零。这一初等论证去除了早期工作中 proper-power 假设，且无需对比值进行一致估计。
英文摘要
We prove that every nonzero integer polynomial with at least two distinct complex roots has lattice point visibility density one, resolving the generalized form of the Visibility Density Conjecture for nonzero polynomials. This extends the origin-passing case established by Chaubey, Pandey, and Regavim. Visibility is taken along the curves $y=tF(x)$ with rational $t$, with a point visible if no positive lattice point on the same curve has a smaller horizontal coordinate. The proof uses a greatest common divisor cutoff to reduce the problem to finitely many equations of the form $F(b)=qF(a)$, with $0<q<1$; for each equation, the positive integers $a$ admitting a positive integer solution $b<a$ form a set of density zero. This elementary argument removes the proper-power hypothesis of earlier work without requiring estimates uniform in the ratio.
发表机构
- Dartmouth College（达特茅斯学院）
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