On the density of visible lattice points along polynomials

TL;DR

This paper explores the density of lattice points visible from the origin along polynomial curves, generalizing previous work on linear and power curves, and introduces a conjecture supported by numerical evidence.

Contribution

It generalizes the concept of visibility to polynomial families, proposes a conjecture on their density, and provides partial results and ideas for proof.

Findings

Established a lower bound for quadratic polynomial families.

Numerical evidence supports the visibility density conjecture.

Discussed potential approaches for proving the conjecture.

Abstract

Recently, the notion of visibility from the origin has been generalized by viewing lattice points through curved lines of sights, where the family of curves considered are $y=mx^k$, $k\in\mathbb{N}$. In this note, we generalize the notion of visible lattice points for a given polynomial family of curves passing through the origin, and study the density of visible lattice points for this family. The density of visible lattice points for family of curves $y=mx^k$, $k\in\mathbb{N}$ is well understood as one has nice arithmetic interpretations in terms of a generalized gcd function, which seems to be absent for general polynomial families. We pose "Visibility density conjecture" regarding the density of visible lattice points for polynomial families passing through the origin, and show some numerical results supporting the conjecture. We obtain a lower bound on the density for a class of quadratic families. In addition, we discuss some ideas of the proof of the conjecture for the simplest quadratic family.