I'm studing the ring of formal series with complex coefficients $\mathbb{C}[[x]]$. I proved that the polynomial $y^2-x^3-x^2$ is irreducible in $\mathbb{C}[x,y]$ but reducible in $\mathbb{C}[[x,y]]$. In fact we can factorize $y^2-x^2(x+1)$. So if we take $u=x\sqrt{x+1}=x(1+1/2x^2-1/8x^3+\cdots)$ we can write $(y-u)(y+u)$.
But sometimes these polynomials can't be factorized in $\mathbb{C}[[x,y]]$, because geometrically are cusps.
Is there a method to classify the polynomials that correspond to ''irreducible series''? If not, could you give some examples?
I’m late, I know, but any way, there is something like a “criteria” on formal power series.
Let $k$ be an algebraically closed field. If $f(X,Y)\in k[[X,Y]]$ is irreducible then its leading form of $f$ is a power of a linear form.
The proof of this fact involves Weierstrass Preparation Theorem and Hensel’s Lemma.
Edit:
For example: $f(X,Y)=f_r(X,Y)+ f_{r+1}(X,Y)+ f_{r+2}(X,Y)+\ldots$, where $f_i$ is a homogeneous polynomial of degree $i$. If $f$ is irreducible then $f_r(X,Y)=(aX+bY)^r$