# Irreducible Polynomials in two complex variables

I am seeking methods to see if a polynomial $f \in O(\mathbb{C^2},0)$ is irreducible. The subject is really new to my and I am studying for myself, for which I don't see about this subject.

would thank them a lot they could provide me some bibliography where to read about this topic

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What sort of "methods", theoretical, algorithms? For what intended applications? You need to say more to get a good answer. –  Bill Dubuque Sep 14 '10 at 0:49
uy! really any method is welcome, Believe me, I would read. I'm reading a book about desingularizacion of vector fields, this book gives a result about how to calculate the multiplicity of intersection of two divisors, the theorem says roughly the following: Let Editing wait.... –  Keinohrhasen Sep 14 '10 at 0:57
Theorem: The intersection of an irreducible local divisor $D_f$ with another effective local divisor $D_g$ is isolated if and only if the germ $g\left( \tau \right)$ is not identically zero, and multiplicity $D_{f}\overset{0}{.}D_{g}$ of this intersection is equal to ther order $ord_{0}\left( g\left( \tau \right) \right)$ I need examples, therefore, need to read a little about the criteria for irreducible polynomials in two complex variables –  Keinohrhasen Sep 14 '10 at 1:06

## 1 Answer

Here's one idea: if $f(x, y)$ is reducible, then the projective closure of $f(x, y) = 0$ has at least two components, which intersect by Bezout's theorem. These intersections are singularities and therefore can be found by setting all partial derivatives to zero. If no singularities exist then $f$ is irreducible.

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thank you very much, I'm chewing the idea –  Keinohrhasen Sep 14 '10 at 3:39