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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.

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1 Answer 1

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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.

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  • $\begingroup$ Thank you very much for your reply. I was doing some digging and seemed to find an algorithm called “bivariate polynomial reduction” that seemed to be heading in the correct direction. Unfortunately though I’m really by no means an expert on this field, so can’t really see if this is directly applicable. Do you think this technique could be applied to these sorts of problems? $\endgroup$ Feb 9, 2023 at 22:07
  • $\begingroup$ @GiulioCrisanti sorry I'm not an expert in this either. Best of luck with your research! $\endgroup$ Feb 11, 2023 at 15:24

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