5 problems

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…

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…

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

Symmetric Strassen additivity conjecture.