I have a large multivariate polynomial of $n$ variables expressed as a product of terms: $$P(x_1, x_2, \dots, x_n) = T_1(x_1,\dots,x_n) \times \dots \times T_k(x_1,\dots,x_n)\ .$$
The maximum degree of any $x_i$ in a given term is one, and the number of terms where any $x_i$ appears is at most two. The problem is just the number of variables/terms, which can be very large.
I would like to know if there would exist an efficient (polynomial-time) way to evaluate the following partial derivative at a given point:
$$\frac{\partial^n P}{\partial x_1\dots\partial x_n} \ \text{?}$$
I've tried automatic differentiation with e.g. Torch but it doesn't scale up so well. The coefficients of the polynomial are complex numbers.
This answer asks a similar question for an arbitrary function: Numerical computation of the $n^{\mathrm {th}}$ derivative of a multivariate function but I haven't found anything useful in the only answer.
I found elements of answer in the following papers:
Multivariate Polynomial Integration and Derivative Are Polynomial Time Inapproximable Unless P=NP∗, Bin Fu
Approximating Multilinear Monomial Coefficients and Maximum Multilinear Monomials in Multivariate Polynomials, Zhixiang Chen, Bin Fu
On testing monomials in multivariate polynomials, Zhixiang Chen, Bin Fu, Yang Liu, Robert Schweller
This problem doesn't seem to have a polynomial-time solution.