10 problems

Comon's conjecture. The Waring rank of a symmetric tensor equals its tensor rank, and the border Waring rank equals the border tensor rank.

Let $f$ be a degree-$d$ homogeneous polynomial, and let $\wr{f}$ and $\bwr{f}$ denote its Waring rank and border Waring rank, respectively. The Waring rank is the minimum number of…

Let $S$ be the polynomial ring in $n$ variables over the complex numbers. For positive integers $k,d,n$, let $\operatorname{rk}_k^\circ(kd,n)$ denote the $k$-rank of a general form…

Let $f,g\in\mathbb{C}[x_1,\ldots,x_n]$ be homogeneous polynomials of degrees $d$ and $d-1$, respectively. Let $u$ be a new variable, and let $\ell[x_1,\ldots,x_n]$ be the set of al…

Let $f\in\mathbb{C}[x_0,\dots,x_n]$ and $g\in\mathbb{C}[y_0,\dots,y_m]$ be polynomials of the same degree $d$, and let $\operatorname{rk}$ denote their Waring rank. Symmetric Stras…

Ikenmeyer's conjecture. Christian Ikenmeyer conjectured that this bound holds for any $r$, which would imply the Ballico--Bernardi conjecture. It is known to hold for $r\leq 5$; th…

Let $q_n$ denote the quadratic form in $n$ variables and let $\operatorname{rk}(q_n^s)$ be its Waring rank. For fixed $n\in\mathbb{N}$, the rank is compared with the generic rank o…

Let $\operatorname{rk}^{\max}_k(kd,2)$ be the least number of $k$-th powers of degree-$d$ forms needed to represent every binary form of degree $kd$. Let $\ell_1$ and $\ell_2$ be n…

The $k$-ary rank conjecture. The upper bound is an equality:

Symmetric Strassen additivity conjecture.