Question

Let $K$ be an algebraic closed field. A polynomial $h \in K[x_1, \ldots , x_n]$ is called multi-affine, if the degree of $h$ in $x_i$ is at most one for all $i$. Let $e_1, \ldots , e_n$ denote the standard basis of $K^n$. Now let $h \in K[x_1, \ldots , x_n]$ be a multi-affine, homogeneous polynomial and let $f=h^N$ for some $N \in \mathbb{N}$. Cleary, $f$ has the following property: For every $1 \leq i \leq n$ and $v \in K^n$, the univariate polynomial $f(t e_i + v) \in K[t]$ is a power of some univariate polymomial of degree at most one. I conjecture, that the converse is also true: Every homogeneous polynomial with this property is some power of a multi-affine polynomial. It seems to me, that there should be an easy proof or counterexample, but I can't see it at the moment.