The characteristic polynomial for a sequence of matrices

The characteristic polynomial of an $r$-tuple $A = (A_1,..., A_r)$ of $n$ by $n$ matrices is the determinant $\mathrm{det}(I + x_1 A_1 + ... + x_r A_r)$. We show that if $r$ is at least $3$ and $A$ is an $r$-tuple of matrices in general position, then up to conjugacy there are only finitely many $r$-tuples of matrices with the same characteristic polynomial as $A$.

This is joint work with Zinovy Reichstein.