Simple criterion to verify that the real zeros are an irreducible algebraic set

Question

Given an irreducible polynomial $p$, its set of real zeros might be a reducible algebraic set. For example: $p=(x^2-1)^2 +(y^2-1)^2$.

Is there a simple sufficient condition on $p$ so that its real zeros will be an irreducible algebraic set? Something in the lines of Eisenstein's criterion, maybe.

Answer 1

Let $p \in \mathbb{R}[x_1, \ldots, x_n]$ be irreducible. If there is a point $v \in \mathbb{R}^n$ such that $p(v)=0$ and sucht that the gradient $\nabla p(v) \neq 0$. Then, by the Artin-Lang-Theorem, the real zeros of $p$ are irreducible in the Zariski topology.