how to solve functional equation?

How to solve polynomial functional equation $P(x,y)=P(\frac{x-y}{2},\frac{y-x}{2})$?

• Given $P(x,y)$ which is a polynomial function, satisfying $P(x,y)=\displaystyle P(\frac{x-y}{2},\frac{y-x}{2})$. Then why should $P(x,y)$ be $\displaystyle\sum^n_{i=0}a_i(x-y)^i$? Is it unique?

• Answer:

  Define $R(t)=P(t/2,-t/2)$. Then $R$ is clearly a polynomial in $t$ (because compositions of polynomials are polynomials), and your functional equation now says that $P(x,y)=R(x-y)$. Substitute $t=x-y$ into the canonical form of the polynomial $R$, and you have your conclusion.