5 problems

Let $f\in\mathcal{R}_{n,d}$ be a homogeneous polynomial, and let its Waring locus be the set of linear forms, viewed as points of $\mathbb{P}((\mathbb{C}^n)^*)$, that occur in a mi…

Let four general forms in three variables have degrees $3,3,3,2$, respectively, and consider their minimal simultaneous Waring decompositions. Conjecture on quadruple decomposition…

Nonemptiness conjecture. For every form $F$, the fertile locus $\mathcal{F}_F$ is nonempty.

Waring-locus additivity conjecture. If $F=\sum_{i=1}^sF_i\in S$ is a degree $d\geq3$ form such that $F_i\in S^{[i]}$ for all $i=1,\ldots,s$, then

Strassen's decomposition conjecture. If $F=\sum_{i=1}^sF_i\in S$ is a degree $d\geq3$ form such that $F_i\in S^{[i]}$ for all $i=1,\ldots,s$, then any minimal Waring decomposition…