Double zeros of a two variable polynomial encoded in a complex function Say I have a two variable polynomial
$$f(x,y)=xy(1-x-y)$$
where $x$ and $y$ are real.
The solutions to $f(x,y)=0$ are the three lines $x=0$, $y=0$ and $x=1-y$.
I am interested in the "double zeroes" of this function (not sure if this is the appropriate terminology), which I call the points where these three lines intersect. These are namely $(0,0)$, $(0,1)$ and $(1,0)$.
The fact that these points are pairs of real values $(a,b)$ made me wonder if there was a better way to encapsulate the double zeroes of this function in terms of complex numbers. For example the function
$$g(z)=z(z-1)(z-i)$$
with $z$ complex, has zeroes at the same points as the "double zeroes" of $f(x,y)$ if one associates $C\to R^2$. $g$ is clearly not unique, but for a holomorphic polynomial I believe that the most general form would have to be of the form
$$g(z)=a\cdot z^{n_1}(z-1)^{n_2}(z-i)^{n_3}$$
with $n_i$ integers.
My question is, is it possible to obtain a polynomial $g(z)$ from the polynomial $f(x,y)$, without knowing the locations of the "double zeroes" of $f$?. For the example above finding the roots of $f$ was easy, but I am working with complicated polynomials for which finding the "double zeroes" directly to construct a function $g(z)$ is very computationally intensive and must be avoided at all costs.
Thank you very much in advance for any help.
 A: What you're talking about are singular points on a plane curve.
These satisfy at least the condition $f=\partial f/\partial x=\partial f/\partial y=0$.
"Double zero" refers, I think, to crunodes.
At any rate, I can't imagine how you would get such a complex polynomial in an easy way from the coefficients of $f$.
I've done a few searches on Google and cannot find any way to relate complex polynomials to the singular points of an affine real plane curve. All I see are articles on the ways to compute singular points.
A quote from "Singular Points of Algebraic Curves" by Sakkalis and Farouki:

Although singular points play a fundamental role in the theory of
algebraic curves (Coolidge 1959, Walker 1978), their algorithmic
identification and analysis is by no means straightforward. For
example, the determination of the multiplicity of a singular point,
its most basic characteristic, is based on ascertaining which of the
higher-order derivatives of the curve equation vanish at that point.
Even when the curve has simple (rational) coefficients, the
coordinates of its singular points will generally be algebraic rather
than rational numbers, and sophisticated methods, requiring the
capabilities of a computer algebra system, must be invoked to reliably
process them.

