Let $G \in \mathbb{Z}[x_1,...., x_n]$ be a degree $d$ homogeneous polynomial. Let $V$ be the hypersurface defined by $G = 0$ and let $N_p$ be the number of geometrically irreducible components of $V$ over $\mathbb{F}_p$ and $D_p$ be the maximum of their degrees.

I want to find $C> 0$ such that $N_p D_p < C$ for all prime $p$. How can I find such $C$? thank you!


1 Answer 1


The degree of a reducible hypersurface is the sum of the degrees of its components, so $N_p \le d$ (with equality iff $G$ splits into $d$ linear factors when reduced modulo $p$), and $D_p \le d$ (with equality iff $G$ hence $V$ is irreducible modulo $p$).

So you can take $C = d^2$, but obviously we cannot have both maxima simultaneously (i.e. $G$ cannot simultaneously split into linear factors and be irreducible for a given $p$) so we should be able to take $C = d(d-1)$ and in fact probably do much better with more careful analysis, but if you just need some bound then even $d^2$ is fine.


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