Determinantal Criteria for Border Rank of High Degree Polynomials

(for an algebraic geometer):
We fix a projective space $\mathbb{P}^n$ and an integer $r$. We are interested in the defining equations of the $r$-th secant variety to the $d$-uple Veronese reembedding of $\mathbb{P}^n$ (i.e. the Veronese variety), where we assume $d$ is sufficiently large, for instance $d \ge 2r$. With these assumptions we prove that the $(r+1)$-minors of a catalecticant matrix with linear entries are sufficient to define the secant variety set-theoretically if and only if the Hilbert scheme parametrising $0$-dimensional Gorenstein subschemes of $\mathbb{P}^n$ of length $r$ is irreducible. In particular, if $n$ is at most $3$ or $r$ is at most $13$, then the minors are sufficient. If $n$ is at least $4$ and $r$ is sufficiently large, then the locus defined by the minors has some additional components. These results motivate introducing cactus varieties, which generalise the secant varieties, and receive a lot of attention recently.

(for a complexity theorist):
Given a homogeneous multivariate polynomial $F$ of degree $d$ in $n$ variables we can construct a sequence of $d-1$ catalecticant matrices whose entries depend linearly on $F$. Let $r$ be the rank of a "middle" matrix in this sequence. The integer $r$ is a lower bound for the border rank of $F$, but in general they are not equal. In our research we assume in addition that the degree of $F$ is sufficiently high ($d \ge 2r$ is enough), and ask if the border rank is equal to $r$. We show that it is equivalent to a deformation problem for certain non-radical ideals in a polynomial ring. In particular, if $r \le 13$, or $F$ depends only on at most $4$ variables, then the border rank is always equal to $r$. Otherwise, if the number of variables is at least $5$, and $r$ is sufficiently large, then there exist polynomials of any degree $d\ge 2r$, for which the rank of all catalecticant matrices is at most $r$, and the border rank is very high.
These results motivate introducing the cactus rank of polynomial, which generalise the border rank, and receives a lot of attention recently.

(joint work with Weronika Buczynska and Joachim Jelisiejew)