A. Lobsenz and T. Phillips, Lattice point visibility along powers of polynomials, arXiv:2604.23050 [math.NT], 2026. This note is a plain-language companion to the paper and to the two that followed it.

The classical picture

Stand at the origin and look out over the integer lattice. A point \((a,b)\) is visible if no other lattice point sits on the segment between you and it — equivalently, if \(\gcd(a,b)=1\). A classical theorem says the visible points have density \(6/\pi^2 = 1/\zeta(2) \approx 0.608\): roughly 61% of the lattice is in view, and the rest is blocked.

Bending the line of sight

Chaubey and Pandey asked what happens when lines of sight are curves. Fix a polynomial \(F \in \mathbb{Z}[x]\). A point \((a,h)\) with positive coordinates lies on the curve \(y = tF(x)\) with \(t = h/F(a)\), and it is *visible along \(F\)* if no positive lattice point on that same curve has a smaller horizontal coordinate. Write \(D(F)\) for the density of visible points in \([1,N]^2\) as \(N \to \infty\); the straight-line case is \(D(x) = 6/\pi^2\).

Their Visibility Density Conjecture predicts a sharp transition: once \(F\) has at least two distinct roots, blocked points should become negligible, and \(D(F) = 1\).

What the paper proves

For every polynomial \(f \in \mathbb{Z}[x]\) of degree at least \(2\) with positive leading coefficient and at least two distinct roots, and every exponent \(m \ge 2\),

\[D\bigl(f^m\bigr) = 1.\]

The proof recasts visibility as a problem about integer points on auxiliary curves and controls those points with Pila's bound. It is quantitative: the number of invisible points satisfies \(\#\mathrm{Invisible}_F(N) \ll_{F,\varepsilon} N^{1+1/\delta_F+\varepsilon}\) for some \(\delta_F \ge 2\). Chaubey, Pandey and Regavim independently established the case of polynomials passing through the origin.

What came next

Two follow-up papers sharpen and complete the picture. Lattice point visibility along powers of quadratic polynomials (with Tristan Phillips, arXiv:2609.05027) pins down the size of the invisible set for powers of quadratics: it has order exactly \(N \log N\) when \(m \ge 3\).

Density one for lattice point visibility along polynomials with at least two distinct roots (arXiv:2609.06309) proves \(D(F) = 1\) for every nonzero integer polynomial with at least two distinct complex roots, resolving the generalized form of the conjecture for nonzero polynomials. The two-root hypothesis is sharp, and the theorem is formally verified in Lean 4.

References

  1. Lobsenz, A. & Phillips, T. Lattice point visibility along powers of polynomials. arXiv:2604.23050 [math.NT], 2026.
  2. Lobsenz, A. & Phillips, T. Lattice point visibility along powers of quadratic polynomials. arXiv:2609.05027 [math.NT], 2026.
  3. Lobsenz, A. Density one for lattice point visibility along polynomials with at least two distinct roots. arXiv:2609.06309 [math.NT], 2026.
  4. Chaubey, S. & Pandey, A. K. On the density of visible lattice points along polynomials. arXiv:2109.08431, 2021.