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I know a homogeneous polynomial $f(x,y)$ is irreducible if and only if $f(x,1)$ is. (Proof?)

I'm wondering if there's a similar criterion to check if $f(x,y)+c$ is irreducible, given that $f$ is homogeneous and irreducible.

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    What do you mea$n$ by irreducible here? How could any monomial with total degree greater than 1 be irreducible? $$x$y^2$ seems like a cou$n$tere$x$ample, at my current (low) level of understanding.2012-09-06

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For your first question, it is only true that if $P(X,1)$ is reducible then $P(X,Y)$ too. (rschwieb gave a counterexample.)

If $P(X,1)$ is reductible, there exist $Q,R \in k[X]$ such that $P(X,1)=Q(X)R(X)$. So $P(X,Y)=Y^d P(X/Y,1)= Y^d Q(X/Y) R(X/Y)$ in $k(X,Y)$. But $d= \text{deg}(P) = \text{deg}(R)+ \text{deg}(Q)$. So there exit $n,m \in \mathbb{N}$ such that $n+m=d$ and $Y^nQ(X/Y) \in k[X,Y]$ and $Y^mR(X/Y) \in k[X,Y]$. Thus, $P(X,Y)$ is reducible.

For your second question, I think there is no such criterium: for example, $XY$ is reducible but $XY+c$ is not for $c \neq 0$, and $X^2$ is reducible and $X^2-c^2$ is reducible too.

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    Exact, it depe$n$ds o$n$ the existence o$f$ a square root. So I added a square to rectify this point. Thank you.2012-09-06