Polynomial in two variables such that $P(x,y)=P(x+y,x-y)$

Question

$P:R\times R \to R\times R$ is a polynomial with real coefficients. It is given that $P\left(x,y\right) = P\left(x+y,x-y\right)$. Find all such $P$.

Answer 1

Hint: Apply $P(x,y)=P(x+y,x-y)$ once more and you get $$ P(x,y)=P(x+y,x-y)=P(2x,2y). $$