The weak cylinder conjecture for polynomial graphs of bounded degree

Ball’s weak cylinder conjecture asserts that a set of $p^2$ points in $\mathrm{A}\mathrm{G}(3,p)$ with at least $p$ non-determined directions is a union of $p$ parallel lines. We prove the conjecture for sets admitting a polynomial graph representation of total degree at most $D$, provided that $p>D^{9\mathrm{/}2}$. More precisely, a non-cylindrical graph of degree $D\ge 3$ satisfies $\mathrm{\mid }{\mathcal{N}}_S\mathrm{\mid }\le p+1-\lceil p\mathrm{/}(D-1)\rceil$ under the same hypothesis. This bound is attained when $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}D-1)$. The proof relates non-determined directions to permutation polynomials on parallel affine lines. A rigidity lemma for moving line restrictions, together with a reconstruction of their compositional components, bounds the number of contributing directions uniformly in $p$. Directions with affine restrictions are treated separately; all other contributions are bounded using geometric indecomposability and the Weil bound for plane curves.