Visibility of lattice points across polynomial curves

Abstract: The visibility of lattice points from the origin along a polynomial family of curves constitutes a significant generalization of visibility along straight lines. Following the classical notion, where the density of visible lattice points equals $1/\zeta(2)$, and its generalization to monomial curves of the form $y = ax^b$, where the density equals $1/(b+1)$, we study a family of polynomial curves defined by $$ y \;=\; q\bigl(a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x\bigr), $$ where $q$ is a positive rational number.

We introduce a new criterion based on a \emph{polynomial greatest common divisor condition} that provides a lower bound on the number of visible lattice points in $\mathbb{N}^2$. Conversely, we derive conditions under which a given lattice point becomes the next visible point along such a polynomial curve. Using the principle of inclusion-exclusion, we obtain an exact double-sum formula for the number of pairs $(a, b) \leq N$ that are visible with respect to this polynomial family. Finally, we extend the framework to related problems and pose several open questions concerning gap distributions and quantitative bounds for non-visible points. This work provides a broader theoretical foundation for lattice point visibility beyond linear and monomial settings.

number theory

Audience: researchers in the topic

( paper )

Combinatorial and additive number theory seminar (CANT 2026)