Defining the domain $$\Gamma[V]=\mathbb C [\bar x ]/I(V)$$ for any irreducible variety $V\subset \mathbb C^n$ (by variety, I mean only zero set of a family of polynomials), $\Gamma(V)$ for the field of fractions of $\Gamma[V]$ and $$\dim V=\operatorname{trdeg}_{\mathbb C}\Gamma (V),$$
(1) I am looking for different proofs that $$\dim V(P)\geqslant n-1$$
for a nonconstant irreducible polynomial $P\in \mathbb C[x_1,\dots,x_n]$.
(2) Why is it harder to show that $\dim V\cap V(P)\geqslant\dim V-1$ for any variety $V$?
One proof of (1)? Since $P$ is nonconstant, $P$ is not algebraic over $\mathbb C$, so there are $P_2,\dots,P_n$ such that $(P,P_2,\dots,P_n)$ is a transcendence basis of $\mathbb C[x_1,\dots,x_n]$ (here we use the fact that the transcendence degree is well defined and in this case equal to $n$, which has a simple, elementary, combinatorial proof here) Now $(P)$ is radical since $P$ is irreducible, so $I(V(P))=(P)$ by the Nullstellensatz, so we are looking for $\operatorname{trdeg}_{\mathbb C} \mathbb C[x_1,\dots,x_n]/(P)$ which is greater than or equal to $\operatorname{trdeg}_{\mathbb C}(P,P_2,\dots,P_n)/(P)=n-1$.