I have several pairs of bivariate polynomials that I want to find if they have common factors. The polynomials, however, have numerical errors because the coefficients are some algebraic numbers that I cannot obtain very easily so I represent them by floating point numbers. Increasing the precision of the coefficient will not work. So one way I do in Mathematica is to just plot the two polynomials on the real plane and see if they have common components or intersect in a tangent. This visual inspection is not ideal since I have several of these polynomials. So I was wondering if one can do such approximation by some form of "epsilon"-GCD in Mathematica. Any ideas? Here is an example of a code I write to check if two such polynomials have common factors:

```
{f, g} = {x0^3 - 8.05690292794*x0^2*x1 + 66.8206602296*x0^2 - 
    2.42037371481*x0*x1^2 + 17.8341472094*x0*x1 - 55.3041827489*x0 + 
    8.52288118623*x1^3 - 66.7864893146*x1^2 - 35.9578468971*x1 + 
    71.4328040275,
   -1.7128257326*x0^2*x1 - 0.309381807141*x0^2 + 
    13.9547615636*x0*x1^2 - 111.58197742*x0*x1 - 20.6540588108*x0 + 
    1.0984988335*x1^3 - 23.4487086797*x1^2 + 116.275997087*x1 + 
    18.7247500823};

ContourPlot[{f == 0, g == 0}, {x0, -100, 100}, {x1, -10, 10}, ContourStyle -> {Blue, Dashed, Thick, Green}]

PolynomialGCD[f, g]

```

You will see that the output of PolynomialGCD will be 1. I want to do this for a lot of pairs of polynomial without visually inspecting them. The plot will give me an idea though that the two polynomials represent curves that share a line

[![enter image description here](https://i.stack.imgur.com/uNL8P.jpg)](https://i.stack.imgur.com/uNL8P.jpg)

Any hints/suggestion is appreciated.