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Given $n \geq 2$, I would like to find either a homogeneous polynomial $F \in \mathbb{Q}[x_1, \ldots, x_n]$ of degree $d > 1$ with the following properties:

  1. $W = \{ \mathbf{x} \in \mathbb{R}^n : F(\mathbf{x}) = 0, \nabla F \not = \mathbf{0} \} \cap (\mathbb{R}_{>0})^n \not = \emptyset$.

  2. For all $\mathbf{z} \in W$, we have $\frac{\partial F}{\partial x_i} \cdot \frac{\partial^2 F}{\partial x_i^2} \leq 0$ for every $i \in \{1, \dots, n\}$.

or a proof that such polynomial $F$ doesn't exist.

I would greatly appreciate if someone could provide me an example of such a polynomial, or an argument on why such polynomials do not exist. My guess is that "most" random choices of polynomial with non-empty $W$ will not satisfy property 2, but I am not really sure. Any helpful comments are appreciated as well. Thank you very much.

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    $\begingroup$ F=0 does the job ;-) $\endgroup$
    – gcousin
    Commented Dec 3, 2017 at 15:52
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    $\begingroup$ @gcousin. More generally, if $W$ is contained in the singular locus of $F$, e.g., for $F=G^2$. Perhaps the OP would like to specify that $W$ is not contained in the singular locus of $F$. $\endgroup$
    – Jason Starr
    Commented Dec 3, 2017 at 15:59
  • $\begingroup$ @JasonStarr Let me fixed this... $\endgroup$
    – Johnny T.
    Commented Dec 3, 2017 at 16:19
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    $\begingroup$ There are still plenty of examples. For instance, fix an integer $d$ with $2\leq d \leq n$, and consider $F(x_1,x_2,\dots,x_n) = \sum_{1\leq i_1<i_2<\dots<i_d \leq n} a_{i_1,i_2,\dots,i_d}x_{i_1}x_{i_2}\cdots x_{i_d},$ i.e., a general linear combination of square-free monomials of total degree $d$. The partial derivatives $\partial^2 F/\partial x_i^2$ are all zero, but the singular set does not equal all of $W$. $\endgroup$
    – Jason Starr
    Commented Dec 3, 2017 at 16:34
  • $\begingroup$ @JasonStarr True. Thanks very much! $\endgroup$
    – Johnny T.
    Commented Dec 3, 2017 at 17:17

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