Let $K$ be an infinite field positive characteristic and $F(X,Y)\in K[[X,Y]]$. Assume that $F(Z\_1+U\_1,Z\_2+U\_2)=F(Z\_1,Z\_2)+F(U\_1,U\_2)$ where $Z\_1,Z\_2,U\_1,U\_2$ are four indeterminates. Can one assert that $F(X,Y)=G(X)+H(Y)$ with $G,H\in K[[X]]$?



Linear formal series in positive characteristic
Yes. It suffices to consider the case that $F$ is a homogeneous polynomial. Write $F(X,Y) = aX^d + Y^eP(X,Y)$, where $d = \deg(F)$ and $P$ is homogeneous of degree $d - e$, $e \geq 1$. 


Claim: if $a \neq 0$, then $d$ is a power of the characteristic $p$ of $K$. 


Proof: Otherwise the coefficient of $Z\_1^{d-1}U\_1$ in $F(Z\_1+U\_1, Z\_2+U\_2)$ is nonzero, whereas this coefficient is zero in $F(Z\_1,Z\_2) + F(U\_1, U\_2)$. 


Now write $F = aX^d + bY^d + Q(X,Y)$, where both $X$ and $Y$ divide $Q$. We will show that $Q = 0$. Indeed, by the claim, $Q$ also satisfies: $Q(Z\_1+U\_1, Z\_2+U\_2) = Q(Z\_1,Z\_2) + Q(U\_1, U\_2)$. Plugging in $Z\_2 = -U\_2$ in the preceding identity yields that $Q(Z\_1,-U\_2) + Q(U\_1, -U\_2) = Q(Z\_1+U\_1, 0) = 0$. If $Q$ is a nonzero polynomial, this gives a contradiction. 


