Waring rank of monomials, and how it depends on the ground field

Question

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.

Answer 1

The answer to question 1 is affirmative. There are several lower bounds in various papers. I'll take the idea from https://arxiv.org/abs/1503.08253 (Buczyński and myself, "Some examples of forms of high rank"). For any homogeneous form $F$, let $\operatorname{Derivs}(F)$ be the vector space spanned by all the partial derivatives of $F$ of all orders (including $F$ itself, as $0$th order). For example $\operatorname{Derivs}(x_1 \dotsm x_d)$ is spanned by all square-free products of $0$ or more variables, so it has dimension $2^d$.

For any homogeneous form $F$, $$ \operatorname{rank}(F) \geq \dim \operatorname{Derivs}(\partial F/\partial x_1) - \operatorname{Derivs}(\partial^2 F / \partial x_1^2) . $$ This is always written over $\mathbb{C}$, but it holds in any field of characteristic $0$ or greater than $\deg(F)$. Also, $\partial / \partial x_1$ can be replaced with any order $1$ partial differential operator, but for simplicity in this answer I'll stick to this.

For $F = x_1\dotsm x_d$, then, we get $$ \operatorname{rank}(x_1\dotsm x_d) \geq \dim \operatorname{Derivs}(x_2 \dotsm x_d) - \dim \operatorname{Derivs}(0) = 2^{d-1} - 0. $$

For question 2 the answer is also affirmative. Observe that the coefficient of $x_1 \dotsm x_d$ in the power $(a_1 x_1 + \dotsb + a_d x_d)^d$ is equal to $$ \binom{d}{1,1,\dotsc,1} \cdot a_1 \dotsm a_d, $$ where the binomial coefficient $\binom{d}{1,1,\dotsc,1}$ is equal to $d!$.

If the characteristic is less than or equal to $d$, this coefficient is $0$ (regardless of what are the $a_i$'s). So $x_1 \dotsm x_d$ is not in the span of $d$'th powers.