Question 46
We call the three variable polynomial $P$ cyclic if $P(x, y, z) = P(y, z, x)$. Prove that cyclic three variable polynomials $P_1$, $P_2$, $P_3$ and $P_4$ exist such that for each cyclic three variable polynomial $P$, there exists a four variable polynomial $Q$ such that
$$
P(x, y, z) = Q(P_1(x, y, z), P_2(x, y, z), P_3(x, y, z), P_4(x, y, z)).
$$
$$
P(x, y, z) = Q(P_1(x, y, z), P_2(x, y, z), P_3(x, y, z), P_4(x, y, z)).
$$