Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Consider a multivariate polynomial map $F:\mathbb{R}^n \rightarrow \mathbb{R}^n.$ Is it always true that the set $C$ in $\mathbb{R}^n$ of critical values is closed?

More specifically, the Sard's theorem tell us that measure of critical values is zero. But when dealing with Polynomials can one easily show that this set is closed nowhere dense?

In other words, does there exists an open dense set S of the real Euclidean space such that for all $c \in S,$ the polynomial system $F(x) -c =0$ has only non-singular roots?

share|improve this question
    
I have the same question about the set of critical values of $f$: $\mathbb{R^n}\rightarrow \mathbb{R}$ is finite. Someone help us, please! Thanks. –  user52523 Dec 14 '12 at 15:38
    
positive dimensional sets can exist –  user64883 Mar 4 '13 at 3:34

Your Answer

 
discard

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

Browse other questions tagged or ask your own question.