Let $k$ be a field and consider the power series $$ f_\pm = (1+y^2)^{1/2}x \pm y \quad\text{ and }\quad g_\pm = x \pm (1 - x^2)^{1/2}y, $$ in the ring $k[[x,y]]$. We can use the $\mathbb N$-"grading" on $k[[x,y]]$ to see that these four elements are irreducible: the degree-one part of $f_\pm$ and $g_\pm$ is $x \pm y$, which is irreducible. Evidently these power series satisfy the relation $$ f_+\cdot f_- = x^2 - y^2 + x^2y^2 = g_+\cdot g_-. $$ Therefore, since $k[[x,y]]$ is a UFD, these two factorizations ought to be the same up to units; that is, it ought to be the case that either
- there are units $u_\pm$ with $f_\pm = u_\pm g_\pm$, or
- there are units $v_\pm$ with $f_\pm = v_\pm g_\mp$.
Question: Which of 1 or 2 is the case? What are $u_\pm$ (or $v_\pm$)? How do you see directly that one of these power series divides the other?
Source: I arrived at this question when I was studying singularities of plane algebraic curves by completing the coordinate ring at the singularity. The factorization above shows that the coordinate ring $k[x,y]/(x^2 - y^2 + x^2y^2)$ of the curve $C : x^2 - y^2 + x^2y^2 = 0$ becomes isomorphic to $k[[s,t]]/(st)$ upon completion at $(x,y)$: we can take $s = f_+$ and $t = f_-$ (or $s = g_+$ and $t=g_-$). So "formally locally", this singularity is the same as that of two lines crossing.
Updates: Lubin showed that only case 1 is possible: multiplying a power series by a unit can only change the monomial term up to a constant factor. (This is obvious, and I shouldn't have missed it!)
The very partialest answer, but:
It’s perfectly clear that it’s Case 1 that holds, since $f_+\equiv x+y\pmod{(x,y)^2}$ and $g_-\equiv x-y\pmod{(x,y)^2}$, and there’s no constant that multiplies $x+y$ into $x-y$.
As for a unit power series $U(x,y)$ for which $\,f_+U=g_+$, you can compute $$ U(x,y)\equiv1-\frac12xy-\frac18(x^3y-x^2y^2-xy^3) -\frac1{16}(x^5y-x^4y^2+x^2y^4+xy^5)\pmod{(x,y)^8}\,, $$ and I assure you that the signs inside the parentheses are correct. I was ready to work out the degree-$8$ terms, but I don’t think that seeing them would help one’s understanding any.
I stared at the situation, but don’t see any plain way of getting the existence of $U$. Maybe someone else with clearer vision can see the way.