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

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.

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.