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I have result: measure of the set of critical values of $f$ is zero (by Sard's theorem), where $f: \mathbb{R^n} \rightarrow \mathbb{R}$ are polynomial functions. How do you show that the set of critical values of $f$ is finite?

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    @copper.hat It isn't obvious from what you wrote. How do reduce the problem to finding the zeros of some single variable polynomial? The critical points which are the zeros of the differential can certainly be infinite, but it is the critical values that are finite.2012-12-16
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    @Matt: You are correct, I didn't see the $n$ on the $\mathbb{R}^n$. It is not as immediate as I thought.2012-12-16
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    Thanks @copper.hat, but I cannot solve this problems. :(2012-12-18
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    Sorry, I was too quick off the mark. I have spent some time looking at this, but have made no headway at all. Do you have any other hints?2012-12-18
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    I have no idea. @Matt Could you give us some idea about this problem?2012-12-18
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    I don't know how to do it. I was just pointing out why the other thing didn't work. I have two "plausibility" arguments that I don't see how to make rigorous. One is that you could view the polynomial as the "height function" of some manifold. The critical values are the heights at which some turning point happens (peaks, valleys, or saddle-like points). It makes sense that a polynomial surface would only have finitely many levels that this happens. Also, you could try to set up some gigantic linear system and find that if you force too many critical values it is inconsistent.2012-12-20
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    Thanks @Matt for sharing your idea.2012-12-20
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    @richard, Could you give us some idea about this problem?2012-12-20
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    @Matt: My mathematical background has some algebraic holes. It there a well-known connection between $\mathbb{R}[x_1,...,x_n]$ and a linear system?2012-12-20
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    All I meant by this was assume for example $f(x,y)=Ax^2+By^2+Cxy+Dx+Ey+F$ has two distinct critical values: $f(x_0, y_0)=G_0$ and $f(x_1, y_1)=G_1$. You get a 6x6 linear system to solve for the coefficients of $f$ using that $Df(x_0, y_0)=0$ and $Df(x_1, y_1)=0$. If you suppose there are too many critical values the system is overdetermined and possibly inconsistent which will show no choice of coefficients could produce a degree $d$ polynomial with a huge number of critical values. This would be really tough to get in full generality though.2012-12-21
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    Thanks @Matt, I have received some hints about this problem: 1, Critical values of f is semi-algebraic set. 2, The semi-algebraic in $\mathbb{R}$ have measure $=0$ then it is finite.2012-12-25
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    @copper.hat: thanks for your kind help.2012-12-26
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    @user52523: Sorry I wasn't of more direct help!2012-12-26
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    Thanks everyone for helping solve this problem. I have found the book: "Real algebraic geometry" of Coste. It contains result about this problem completely. Thanks @copper.hat: you helped me so much. So does Matt.2012-12-30
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    Wow, well done! Persistence is the key to success!2012-12-30

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