Polynomial on Unique Factorization Domain

Question

Let $\mathbf{A}$ be a unique factorization domain and $S\in\mathbf{A}[x_1,\cdots,x_r,y_1,\cdots,y_s]$ be a homogeneous polynomial such that $S(x_1,\cdots,x_r,0,\cdots,0)$ is irreducible over $\mathbf{A}$. How can I show that $S$ is also irreducible over $\mathbf{A}$

Answer 1

Suppose we have that $S(x,y)=A(x,y)B(x,y)$ where $x=(x_1,\cdots,x_r)$ and $y=(y_1,\cdots,y_s).$ Both $A$ and $B$ are also homogeneous and WLOG we assume that $B(x,0)$ is a unit since $S(x,0)$ is irreducible.

Now, since $B(x,0)$ is non-zero it must have terms which contain no $y_i$, but since it's a unit these terms must be a unit $A$. However, $B(x,y)$ is homogeneous and so we conclude that it has degree $0$ and so is in fact a unit as well. Hence we conclude that $S(x,y)$ is irreducible.