$\newcommand\C{\mathbb{C}}$ $\newcommand\bs{\backslash}$ I have several questions which are related to one question but for which I slightly change the original question. I believe the answer to all is yes, but I have not worked any of them out:
Let $f(x,y)\in \C[x,y]\bs\C$ be an arbitrary bivariate non-constant polynomial. Can we find a perturbation of the constant term such that $f$ can be changed to an irreducible polynomial, i.e. can we find $\epsilon \in \C^*$ such that $f(x,y)+\epsilon$ is irreducible.
Can we do the above assuming $\epsilon\in\mathbb R^*$.
Can we do 1.) for any field $K$ of any characteristic
Can we do 1.) for any non-constant multivariate (not univariate) polynomial for any field of any characteristic?
Note: to clarify I assume in 1) $f$ is bivariate and not univariate. So in fact, $f\in \C[x,y]\bs(\C[x]\cup\C[y])$