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.


1 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.