Let $A$ be an $n \times n$ matrix s.t. every entry in $A$ is either $0$ or polynomially-bounded. That is, there exists a constant $c$ for which $n^{-c }\le A[i,j] \le n^{c}$.
Let $A = VJV^{-1}$ be a Jordan decomposition where the columns of $V$ have norm $1$. I am interested in upper-bounding $\|V^{-1}\|$, even very crudely (say, at most $n^{n^{d}}$ for some constant $d$).
I trying going through $\mathrm{det}(V^{T}V)$, for which an upper bound is easy, but could not come up with a lower bound.