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](https://www.sciencedirect.com/science/article/pii/S0021869311004728) 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.



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


