Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.