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

Some examples of $n$-exceptions are:

1. $\quad p_3(x_1,\ldots,x_n)=x_{1}^{3}+\cdots+x_{n}^{3}\ $ is an $n$-exception, for every $n\geq 3$.

In general, the homogeneous symmetric polynomials: $$p_{k}(x_1,\ldots,x_n)=x_{1}^{k}+\cdots+x_{n}^{k}$$ $$e_{k}(x_1,\ldots,x_n)=\sum_{1\leq i_1<\cdots<i_k\leq n}x_{i_1}\cdots x_{i_k}$$ are both $n$-exception if $n\geq 3$ and $k>1$. Let's say that these examples are trivial cases of $n$-exceptions.

Some non trivial cases are the following:

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.

Notice that $p_{1}^{d}$ is not an $n$-exception for every $n$ and $d$. Also there are homogeneous symmetric polynomials that are not $n$-exceptions for any $n\geq 3$ like $m_{21}$ and $h_3$.

It is known that $\dim_{\mathbb{R}}(\langle{A_f}\rangle)\geq n$ when $d=3$ and $f$ is not a scalar multiple of $p_{1}^{3}$. There is a conjecture that the last inequality holds also for any degree $d\geq 4$ (see in mathoverflow: Dimension of the span of all partial derivatives of a given homogeneous symmetric polynomial $f$ and the polynomial $E(f)$)

What I tried is to show first the existence of $n$-exceptions for any degree $d$. When $d=3$ they are completely characterized already:

Let $n\geq 3$ and $f=a\,m_{3}+b\,m_{2,1}+c\,m_{1,1,1}$ (as a linear combination of the monomial basis, with $a,b,c\in\mathbb{R}$) is an $n$-exception if and only if $$6a(2b+(n-2)c)=4(n-1)b^2.$$ If $n=2$, the condition is $b=0$ or $b=3a$.

The problem is when $d\geq 4$ I only have verified the following observation:

1. For $3\leq d\leq 6$, the symmetric polynomial $p_{2}p_{1}^{d-2}$ is a $(d+1)$-exception.

and I tried to use the following fact:

2. Any homogeneous polynomial $f$ of degree $d$ is a linear combination of $k$, $d$-th powers of linear forms, for some natural $k$.

My question: How we can start to characterize $n$-exceptions for degree $d\geq 4$ in general in terms of it's coefficients in the monomial basis or other basis of the ring of symmetric functions ?

Any idea on this is very welcome, Thank you!