I am curious to know if any given $n$-variable polynomial in $\mathbb{R}[\mathbf{x}]$, not in standard form, can be simplified by an algorithm in polynomial time. The polynomial is $$ p(\mathbf{x}) = \sum \limits_{i=1}^{m}p_i(\mathbf{x})^{k_i}.$$ The entries of the problem are:
$n \rightarrow \text{number of variables};$
$d_i \rightarrow \text{degree of each polynomial $p_i(\mathbf{x})$}$;
$ p_i(\mathbf{x}) \rightarrow \text{random polynomials of degree $d_i$};$
$m \rightarrow \text{number of random polynomials}$;
$k_i \rightarrow \text{positive integer}.$
For better understanding, let the polynomial
$$\small p(x_1, x_2, x_3) = (x_1 + x_2x_3)^3 + (x_2+x_3 +3)^2 + (x_1-x_2 +1).$$ It can be simplified in its standard form as
$$\small p(x_1,x_2,x_3) = x_2^3x_3^3 + 3x_1x_2^2x_3^2+3x_1^2x_2x_3 + x_1^3 +x_3^2 +x_2^2 +2x_2x_3+6x_3 +6x_2+x_1 -x_2+10.$$
If this problem cannot be solved in polynomial time, what is its complexity class?
Remark: In the standard form of an $n$-variable polynomial $p(\mathbf{x}) = \sum_{\mathbf{\alpha}}{p_{\mathbf{\alpha}}x^{\mathbf{\alpha}}}$, with $\mathbf{\alpha} \in \mathbb{N}^{n}$, all monomials $x^{\mathbf{\alpha}}$ are unique.
Thanks in advance!
Remember that for every polynomial $p(x)$
$$p(x)=\sum \frac{f^{(\alpha)}(0)}{\alpha!}x^\alpha$$
Here $x=(x_1,x_2,...,x_n)$, $\alpha=(\alpha_1,\alpha_2,...,\alpha_n)$, $f^{(\alpha)}=\frac{\partial^{\alpha_1+...+\alpha_n}}{\partial x_1^{\alpha_1}...\partial x_n^{\alpha_n}}f$, $x^{\alpha}=x_1^{\alpha_1}...x_n^{\alpha_n}$, and $\alpha!=\alpha_1!...\alpha_n!$.
So, as long as you can compute the $f^{(\alpha)}(0)$ in polynomial time we are done.
But how do we (/can we) compute derivatives of a polynomial expression as the one you wrote?