Representing FP relations using witnesses
An $A$-polynomial relation can't have two many pairs agreeing on the first coordinate. Namely for any $x$ there can be at most $2^{p(x)}$ $y$'s such that $(x,y)\in R$. This shows that the simple relation $R=\{0,1\}^*\times\{0,1\}^*$ is in FP but not A-polynomial.