Lattice point visibility along powers of quadratic polynomials

Abstract: We study the growth of the number of invisible lattice points along powers of quadratic polynomials. Let $f(x)=Ax<sup>2+Bx+C\in\mathbb{Z}[x]$ have a positive leading coefficient and nonzero discriminant, and let $F(x)=f(x)<sup>m$ with $m\geq 2$. For $m\geq 3$ we prove that the number of invisible lattice points in $[1,N]<sup>2$ has order $N\log N$, and when $m=2$ the number of invisible lattice points satisfies $N\log N \ll_F#\mathrm{Invisible}_F(N)\ll_F N(\log N)<sup>4$. These estimates refine a previous result of the authors.