The Waring rank of a degree-$d$ homogeneous polynomial $p$ is the least integer $r$ such that you can write $p$ as a linear combination of $r$ $d$-th powers of linear forms $\{\ell_k\}$: $$ p = \sum_{k=1}^r c_k \ell_k^d $$ for some scalars $\{c_k\}$. Lets use $\operatorname{rank}(p)$ to denote the Waring rank of $p$.

If the ground field does not have characteristic $2$ then, for example, $\operatorname{rank}(xy) = 2$, since we can write $$ xy = \frac{1}{4}\left((x+y)^2 - (x-y)^2\right), $$ but we cannot write $xy$ as a scalar multiple of just a single square of a linear form.

Similar decompositions of the monomial $x_1x_2\cdots x_d$ show that $\operatorname{rank}(x_1x_2\cdots x_d) \leq 2^{d-1}$ as long as the ground field has characteristic $0$ or strictly greater than $d$, and equality holds if the ground field is $\mathbb{C}$.

**Question 1:** Is it true that $\operatorname{rank}(x_1x_2\cdots x_d) = 2^{d-1}$ for any field that has characteristic $0$ or strictly greater than $d$?

**Question 2:** Is it true that $\operatorname{rank}(x_1x_2\cdots x_d) = \infty$ (i.e., $x_1x_2\cdots x_d$ cannot be written as a linear combination of $d$-th powers of linear forms) if the ground field has characteristic between $2$ and $d$ (inclusive)? I can show that this is true for $d = 2$, but I am not sure about the general case.