Density one for lattice point visibility along polynomials with at least two distinct roots

Abstract

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.