# Number of geometrially irreducible components of hypersurface over $\mathbb{F}_p$

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!

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.