Solution mod 2 of polynomial equation

Question

Can one find a polynomial $p(x,y)$ such that it is integer for integer $x,y$ and it satisfies $$ p(x,y) + p(y,x) = x^2 + y^2 +1 \mod 2$$ or prove that it is not possible?

Answer 1

Substitute $x=y=0$. Looks like you want $2p(0,0)\equiv1\pmod2$. This is not possible, if $p(0,0)$ is an integer.

Answer 2

No; simply set $x=y$. Then the LHS is $2p(x,x) \equiv 0 \pmod 2$, and the RHS is $2x^2 + 1 \equiv 1 \pmod 2$.