Fix two vectors $\mathbf u$ and $\mathbf v$ in $\mathbb R^k$, and let $\circ$ denote the coordinate-wise Hadamard / Schur product, i.e. $\mathbf u\circ\mathbf v$ has coordinates $w_i=u_iv_i$. Write $\mathbf u^{\circ r}$ for the $r$th power.
Let $r>1$ be a given number. Consider the subset $S=\{(\kappa \mathbf u +\lambda\mathbf v)^{\circ r}\}_{(\kappa,\lambda)\in\mathbb R^2}$ of $\mathbb R^k$. Now if $r$ is natural, expand to $\kappa^p\lambda^{r-p}\mathbf w_{p+1}$, where the $\mathbf w_p$ may very well be linearly independent and yield $r+1$-dimensional span.
But what if $r>1$ is not integer – what is then the dimension of the span?
If it matters: The cases that led me to this problem, all have rational $r=m/n$ with $m$ odd.
So what I am curious about, would be properties like (I) the dimensionality $d=d(r)$ for the general case (large $k$ and worst-case $\mathbf u$ and $\mathbf v$), (II) a characterization of the degeneracies (beyond the obvious $\mathbf u=c\mathbf 1$ or $=c\mathbf v$ ...), (III) any nice formula for an orthogonal basis.
(I am also curious about whether I manage to get this right – this is my delurking post.)