The Waring rank of a degree-d multivariate polynomial F(x_1,... x_n) is the least number $r$ such that,
F(x_1,... x_n) = c_1\ell_1 + c_2\ell_2 + ... + c_r\ell_r
where \ell_i = \alpha_{i1}x_1 + \alpha_{i2}x_2 + ... \alpha_{in}x_n.
It follows from a simple application (due to Kayal) of the classical partial derivative method of Nisan and Wigderson that
the multilinear monomial $\prod_{i=1}^n x_i$ has Waring rank at least $2^n/\sqrt{n}$ and a result of Fischer gives an upper bound of $2^{n-1}$
A result due to Ranestad and Schreyer (2011) proves that the Waring rank of the multilinear monomial is exactly $2^{n-1}$. A follow up work of
Carlini, Catalisano and Geramita (2012) shows that the Waring rank of a general monomial $\prod_{i=1}^n x_i^{d_i}$ is exactly $\prod_{i=2}^n (d_i + 1)$.
Both these proofs crucially make use of the notion of Apolarity, a notion that dates back to Sylvester. We provide an alternate proof of both these results
and discuss a few natural related questions that remain unresolved.
Part of ongoing work with Nutan Limaye, Sai Sandeep, Srikanth Srinivasan and Thomas Thierauf. |