I've been struggling to understand why the interactive proof for #SAT stops after only $m$ rounds, where $m$ is the number of variables in the formula $\phi$. I understand that two polynomials of degree $d$ can agree for at most $d$ values. As the proof in the Sipser text goes (which I believe is the standard proof), the degree of the polynomial is at most $n$. Why can't the verifier simply check $n+1$ points and prove with certainty whether $\langle\phi, k\rangle \in$ #SAT?
Would this require exponential time? I can't see how the Prover can escape the testing of $n+1$ points.