An arithmetic integrability result for codimension-one foliations on complex projective spaces

Let $\text{F}$ be a codimension-one holomorphic foliation of degree $d$ on ${\text{PP}}^n$, $n\ge 3$, admitting an invariant hyperplane $H$. We study the extremal situation in which $S=(H\cap \text{Sing}(\text{F}){)}_{\mathrm{r}\mathrm{e}\mathrm{d}}$ is an irreducible hypersurface of $H$ of degree $d+1$. When $d+1$ is a power of a prime, we prove that, in suitable homogeneous coordinates with $H=(t=0)$, $$\mathrm{\Omega }=Qdt-\frac{t}{d+1}dQ,$$ where $Q$ is homogeneous of degree $d+1$. Thus $Q\mathrm{/}{t}^{d+1}$ is a rational first integral. The proof reduces the Frobenius equation to a twisted closedness equation on a plane section and uses Zariski’s theorem on the Alexander polynomial of an irreducible plane curve. We also prove a complementary rigidity theorem for an arbitrary smooth invariant hypersurface $D\subset {\text{PP}}^n$: if the reduced singular divisor on $D$ is smooth, irreducible, and of maximal degree, then the same normal-form phenomenon holds, with no arithmetic hypothesis on its degree; in the low-weight range the smoothness assumption on the singular divisor can be dropped. Finally, we show that the principal hypotheses are sharp. Dropping the maximal-degree condition yields a family with irreducible reduced singular support and no non-constant rational first integral. For every $d+1$ which is not a prime power we construct a global counterexample with irreducible maximal-degree singular support, and a final family shows that irreducibility of the reduced support is also genuinely necessary.