Visibility of Lattice Points across Polynomials

TL;DR

This paper generalizes the concept of lattice point visibility from straight lines to polynomial curves, introducing new criteria and formulas to analyze visibility patterns and distributions in this broader setting.

Contribution

It introduces a polynomial gcd-based criterion for visibility, derives exact formulas for counting visible points, and extends the framework to related problems beyond linear cases.

Findings

Established a lower bound on visible lattice points using gcd conditions

Derived an exact double-sum formula for counting visible pairs

Extended the framework to non-linear polynomial curves and posed open questions

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 equals 1/2, and its generalization to monomial curves of the form y = a x^b, where the density equals 1/(b+1), we study a family of polynomial curves defined by y = q(a_n x^n + ... + a_1 x), where q is a positive rational number. We introduce a new criterion based on a polynomial greatest common divisor condition that provides a lower bound on the number of visible lattice points in 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 also obtain an exact double-sum formula for the number of pairs (a, b) less than or equal to 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.