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We say that an homogeneous symmetric polynomial $f(x_1,\ldots,x_n)$ of degree $d$ is an $n$-exception if the dimension of the $\mathbb{R}$-span of the following set of polynomials of degree $d-1$ $$A_f=\left\{\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n},\sum_{j=1}^{n}x_{j}\frac{\partial^2 f}{\partial x_{j}^2}\right\}$$ is equal to $n$.

  In symbols: $$f\ \ \text{is an}\ n\text{-exception}\ \Longleftrightarrow \dim_{\mathbb{R}}\left(\langle{A_f}\rangle\right)=n$$

We say that an homogeneous symmetric polynomial $f(x_1,\ldots,x_n)$ of degree $d$ is an $n$-exception if the dimension of the $\mathbb{R}$-span of the following set of polynomials of degree $d-1$ $$A_f=\left\{\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n},\sum_{j=1}^{n}x_{j}\frac{\partial^2 f}{\partial x_{j}^2}\right\}$$ is equal to $n$. In symbols: $$f\ \ \text{is an}\ n\text{-exception}\ \Longleftrightarrow \dim_{\mathbb{R}}\left(\langle{A_f}\rangle\right)=n$$

It is an current problem to show that this dimension is at least $n$ if $f$ is not a scalar multiple of $p_{1}^{d}=(x_1+\cdots+x_n)^{d}$. In others words: $$f\neq k\,(x_1+\cdots+x_n)^{d},\ \text{for every}\ k\in\mathbb{R} \Longrightarrow \dim_{\mathbb{R}}(A_f)\geq n$$ see here: Dimension of the span of all partial derivatives of a given homogeneous symmetric polynomial $f$ and the polynomial $E(f)$

We say that an homogeneous symmetric polynomial $f(x_1,\ldots,x_n)$ of degree $d$ is an $n$-exception if the dimension of the $\mathbb{R}$-span of the following set of polynomials of degree $d-1$ $$A_f=\left\{\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n},\sum_{j=1}^{n}x_{j}\frac{\partial^2 f}{\partial x_{j}^2}\right\}$$ is equal to $n$.

(Also in Mathematics Stack Exchange: https://math.stackexchange.com/questions/2528000/characterizing-n-exceptions-on-the-ring-of-symmetric-polynomials)

We say that an homogeneous symmetric polynomial $f(x_1,\ldots,x_n)$ of degree $d$ is an $n$-exception if the dimension of the $\mathbb{R}$-span of the following set of polynomials of degree $d-1$ $$A_f=\left\{\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n},\sum_{j=1}^{n}x_{j}\frac{\partial^2 f}{\partial x_{j}^2}\right\}$$ is equal to $n$.

2. $\quad p_{2,1,1}(x_1,\ldots,x_5)\ $ is a $5$-exception.

3. $\quad e_3(x_1,\ldots,x_n)\ $ is an $n$-exception, for every $n\geq 2$.

4. $\quad 9m_{3}+21m_{2,1}+28m_{1,1,1}\ $ is a $4$-exception.

5. $\quad 5m_{4}+14m_{31}+21m_{22}+28m_{211}+35m_{1111}\ $ is a $11$-exception.

6. $\ 2\,m_{3}-3\,m_{2,1}+12\,m_{1,1,1}$ is a $3$-exception

7. $\ 3\,m_{3}+3\,m_{2,1}-2\,m_{1,1,1}$ is a $3$-exception

8. $\ 1\,m_{3}-1\,m_{2,1}+2\,m_{1,1,1}$ is a $4$-exception

9. $\ 16\,m_{3}-12\,m_{2,1}+21\,m_{1,1,1}$ is a $4$-exception

10. $\ 4\,m_{3}-3\,m_{2,1}+4\,m_{1,1,1}$ is a $5$-exception

11. $\ 5\,m_{3}-3\,m_{2,1}+3\,m_{1,1,1}$ is a $6$-exception

12. $\ 10\,m_{3}-5\,m_{2,1}+4\,m_{1,1,1}$ is a $7$-exception

2. $\quad p_{2,1,1}(x_1,\ldots,x_5)\ $ is a $5$-exception.

3. $\quad e_3(x_1,\ldots,x_n)\ $ is an $n$-exception, for every $n\geq 2$.

4. $\quad 9m_{3}+21m_{2,1}+28m_{1,1,1}\ $ is a $4$-exception.

5. $\quad 5m_{4}+14m_{31}+21m_{22}+28m_{211}+35m_{1111}\ $ is a $11$-exception.

6. $\quad 2\,m_{3}-3\,m_{2,1}+12\,m_{1,1,1}\ $ is a $3$-exception

7. $\quad 3\,m_{3}+3\,m_{2,1}-2\,m_{1,1,1}\ $ is a $3$-exception

8. $\quad 1\,m_{3}-1\,m_{2,1}+2\,m_{1,1,1}\ $ is a $4$-exception

9. $\quad 16\,m_{3}-12\,m_{2,1}+21\,m_{1,1,1}\ $ is a $4$-exception

10. $\quad 4\,m_{3}-3\,m_{2,1}+4\,m_{1,1,1}\ $ is a $5$-exception

11. $\quad 5\,m_{3}-3\,m_{2,1}+3\,m_{1,1,1}\ $ is a $6$-exception

12. $\quad 10\,m_{3}-5\,m_{2,1}+4\,m_{1,1,1}\ $ is a $7$-exception