Upper bounds for the rank of powers of quadrics

We establish an upper bound for the rank of every power of an arbitrary quadratic form. Specifically, for any $s\in \mathbb{N}$, we prove that the $s$-th power of a quadratic form of rank $n$ grows as $n^s$. Furthermore, we demonstrate that its rank is subgeneric for all $n>(2s-1{)}^2$.