Find the polynomial functions $f$ with real coefficients such that for every $ A, B \in M_4( \mathbb{R})$ with $\mbox{rank}(A) > \mbox{rank}(B) $ we have that $\mbox{rank}(f(A)) \geq \mbox{rank}(f(B)) $.
How to start? I tried with some examples but doesn't work.