I usually find it difficult to check irreducibility of polynomials in $K[[X,Y]]$ ($K$ algebraically closed). Does anyone know about generic methods that can be used ? And especially of ones that can be applied to polynomials like $XY-(X+Y)(X^2+Y^2)$ (I found it reducible by using Hensel's lemma after a suitable change of variable, if I did not make mistake).



Generic methods to check irreducibility of polynomials in $K[[X,Y]]$
I have a paper giving an irreducibility test (and factoring "algorithm", in some sense of the word) for formal power series over a PID, which should eventually appear in Trans. of the AMS. In particular it applies to $K[[X]][[Y]]$.


See Theorem 1 of <http://arxiv.org/abs/1107.4860>


In essence: Write any polynomial $f$ in $K[[X,Y]]$ as a polynomial in $Y$ with coefficients in $K[[X]]$, and let $f\_0$ and $f\_1$ be the coefficients of $Y^0$ and $Y^1$, respectively. Then $f$ is irreducible in $K[[X,Y]]$ if and only if either (1) $f\_0 = 0$ and $f\_1$ is a unit in $R[[X]]$, or (2) $f$ has a unique factor in $K[[X]][Y]$ with constant term not a unit in $K[[X]]$. (This works, by the way, even if $f$ is in $K[[X]][Y]$.)


Added on 4/3/14: The criterion above is equivalent to the following. A polynomial $f \in K[[Y]][X]$ is irreducible in $K[[X,Y]] = K[[Y]][[X]]$ if and only if one of the following conditions holds.


(1) $f\_0 = 0$ and $f\_{1,0} \neq 0$.


(2) $f\_0 \neq 0$ and $f\_{0,0} = 0$, and if $f = gh$ with $g,h \in k[[Y]][X]$, then either $g\_{0,0} \neq 0$ or $h\_{0,0} \neq 0$.


Here, $f\_{i,j}$ denotes the coefficient of $X^i Y^j$ in $f$, and $f\_i$ denotes the coefficient of $X^i$ in $f$ as a polynomial in $K[[Y]][X]$. To test whether (2) holds or not in a given situation, you need to be able to factor polynomials over the complete DVR $K[[Y]]$. To do so you can use Hensel's lemma, Newton polygons, and other methods for factoring polynomials over local fields.


