Let $Q_1,\ldots,Q_s \in k[x_1,\ldots,x_n]$, where $k$ is not necessarily algebraically closed (I'm thinking of $k$ as some field with positive characteristic $p$). I'm somewhat new to the world of classical algebraic geometry, so the following may be trivial questions:
Let $V = V(Q_1,\ldots,Q_s)$. Decompose $V = V_1 \cup \cdots \cup V_m$, where each of the $V_i$ are irreducible.
- Is there a bound on the number of irreducible components, $m$, that depends on the degrees of the $Q_j$'s?
- Let each of the $V_i$'s be defined by polynomials $f_{i1},f_{i2},\ldots$. If the the original $Q_j$'s are of low degree, can we bound the degrees of the $f_{ik}$'s?
Thanks!