Question 1
A polynomial $P(x,y)$ satisfies, for every integer $n\ge0$, that each of $P(n,y),P(x,n)$ is either the zero polynomial or has degree at most n. Can $P(x,x)$ have odd degree? Both the book and official original include n=0. The book says “a nonzero constant” where the official original says “the zero polynomial”; the effect of this wording is discussed below.