TITLE: Irreducibility after substitution
QUESTION [2 upvotes]: I would like to show that when $f(x,y)$ is irreducible over $\mathbb{C}[x,y]$ then $f(x^2,y)$ is irreducible over $\mathbb{C}[x,y]$.  I know that this is not true in general, for example, $f(x,y) = y^2 - x$.  However, I know that the original polynomial contains only even powers of $x$.  Does this extra condition allow me to conclude $f(x^2,y)$ is irreducible over $\mathbb{C}[x,y]$?

REPLY [0 votes]: Maybe the paper http://math.univ-lille1.fr/~pde/ptmy.pdf by Pierre Debes is relevant. In particular, see Lemma 0.1.
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