Given $n \geq 2$, let $F \in \mathbb{Q}[x_1, \ldots, x_n]$ be a homogeneous polynomial of degree $d > 1$. I would like to find an example (or a proof that such $F$ doesn't exist) 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 $i \in \{1, \dots, n\}$.

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.

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.

Let $F \in \mathbb{Q}[x_1, \ldots, x_n]$ be a homogeneous polynomial of degree $d > 1$ and $n \geq 2$. I am interested in finding an example (or maybe that such $F$ doesn't exist?) 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 $i \in \{1, ..., n\}$.

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

Given $n \geq 2$, let $F \in \mathbb{Q}[x_1, \ldots, x_n]$ be a homogeneous polynomial of degree $d > 1$. I would like to find an example (or a proof that such $F$ doesn't exist) 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 $i \in \{1, \dots, n\}$.

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.

Let $F \in \mathbb{Q}[x_1, \ldots, x_n]$ be a homogeneous polynomial. I am interested in finding an example (or maybe that such $F$ doesn't exist?) with the following properties.

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

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 $i \in \{1, ..., n\}$.

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

Let $F \in \mathbb{Q}[x_1, \ldots, x_n]$ be a homogeneous polynomial of degree $d > 1$ and $n \geq 2$. I am interested in finding an example (or maybe that such $F$ doesn't exist?) 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 $i \in \{1, ..., n\}$.

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