I need some help on the problem below.

Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set

$$ E(f):=\sum_{j=1}^{n}x_{j}\frac{\partial^2 f}{\partial x_{j}^2}. $$

Question: If $f\neq (x_1+\ldots+x_n)^d$ then the dimension of the real span of the set below $$ \big\{\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n},E(f)\big\} $$ is at least $n$. In symbols: $$ \dim_{\mathbb{R}}\left(\mathrm{span}\big\{\frac{\partial f}{\partial x_1},\ldots,\frac{\partial f}{\partial x_n},E(f)\big\}\right)\geq n $$

For degree 3 this assertion is already settled.

Have any one any idea to prove this statement in general ?

  • $\begingroup$ Recall that an equivalent condition for $f$ to be homogeneous of degree $d$ is that $f$ must satisfy identity below $$ \sum_{j=1}^{n}x_{j}\frac{\partial f}{\partial x_j}=d\cdot f. $$ $\endgroup$ May 4, 2015 at 19:03
  • $\begingroup$ Another thing that can be useful to remember here is that any homogeneous polynomial $f$ of degree $d$ is a linear combination of $k$, $d$-th powers of linear forms, for some $k$. $\endgroup$ May 4, 2015 at 19:06


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