How find all $f(x+y,y-x)=f(x,y)$

Question

let the $f(x,y)$ be Polynomial, such $$f(x+y,y-x)=f(x,y)$$ Find all $f(x,y)$

My idea: let $x+y=u, y-x=v$ then $$y=\dfrac{u+v}{2},x=\dfrac{u-v}{2}$$

Answer 1

Only constants can be polynomial solutions.

Let $A(x,y)=(x+y,y-x)$. Then $A(A(x,y))=(2y,-2x)$, so $A(A(A(A(x,y))))=(-4x,-4y)$. We therefore have $f(x,y)=f(-4x,-4y)$, so $f(x,y)=f(16x,16y)$ and hence $f(x,y)=f(16^n x,16^n y)$ for any $n\geq 0$.

The polynomial $Q(t)=f(tx,ty)-f(x,y)$ has infinitely many roots, so $Q$ must be zero. Decomposing $f$ as a sum of homogeneous polynomials, $f=\sum_{k=0}^d f_k$ where the homogeneous degree of $f_k$ is $k$ and $f_d\neq 0$, it is easy to see that the degree of $Q$ is $d$ whenever $d$ is nonzero, so $d$ must be zero.

In other words, $d=0$ and $f$ is constant.