Problem 1775

First we find all polynomials $P(x, y)$ with complex coefficients which satisfies the condition of the problem statement. The identically zero polynomial clearly satisfies the condition. Let consider other polynomials. Let $i^{2}=-1$ and $P(x, y)=(x+i y)^{n}(x-i y)^{m} Q(x, y)$, where $n$ and $m$ are non-negative integers and $Q(x, y)$ is a polynomial with complex coefficients such that it is not divisible neither by $x+i y$ nor $x-i y$. By the problem statement we have $Q(x, y) Q(z, t)=Q(x z-y t, x t+y z)$. Note that $z=t=0$ gives $Q(x, y) Q(0,0)=Q(0,0)$. If $Q(0,0) \neq 0$, then $Q(x, y)=1$ for all $x$ and $y$. Thus $P(x, y)=(x+i y)^{n}(x-i y)^{m}$. Now consider the case when $Q(0,0)=0$. Let $x=i y$ and $z=-i t$. We have $Q(i y, y) Q(-i t, t)=Q(0,0)=0$ for all $y$ and $t$. Since $Q(x, y)$ is not divisible by $x-i y, Q(i y, y)$ is not identically zero and since $Q(x, y)$ is not divisible by $x+i y$, $Q(-i t, t)$ is not identically zero. Thus there exist $y$ and $t$ such that $Q(i y, y) \neq 0$ and $Q(-i t, t) \neq 0$ which is impossible because $Q(i y, y) Q(-i t, t)=0$ for all $y$ and $t$. Finally, $P(x, y)$ polynomials with complex coefficients which satisfies the condition of the problem statement are $P(x, y)=0$ and $P(x, y)=(x+i y)^{n}(x-i y)^{n}$. It is clear that if $n \neq m$, then $P(x, y)=(x+i y)^{n}(x-i y)^{m}$ cannot be polynomial with real coefficients. So we need to require $n=m$, and for this case $P(x, y)=(x+i y)^{n}(x-i y)^{n}=\left(x^{2}+y^{2}\right)^{n}$.