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Hi, I'm new here, and not a mathematician at all :-(

I am looking for an algorithm to find the roots (in the complex domain) of a polynom of several variables.

Thanks for any light you could bring to me.

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The study zeros of polynomials is a vast and sophisticated subject called algebraic geometry. Usually the set of zeros is a curve or surface of some sort, called a variety (en.wikipedia.org/wiki/Algebraic_variety). One of the basic insights of algebraic geometry is that varieties correspond to algebraic objects called ideals (see the same wiki page). In practice these algebraic objects are what is used to perform computations on varieties... –  Steve Huntsman Jun 18 '10 at 17:40
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...If you truly need to learn about this stuff in any generality at all, a good way to start would be (to first acquaint yourself with the basic notions of abstract algebra like [groups], rings, fields, and ideals and then) check out this book: books.google.com/books?id=7eLkq0wQytAC –  Steve Huntsman Jun 18 '10 at 17:42
    
I second Steve Huntsman's book recommendation. I'm afraid the question as it stands is a bit broad for MathOverflow - since the set of roots forms a complex hypersurface, it is difficult to algorithmically "find" all of them, except by descriptive means (e.g., using the polynomial itself, or computing interesting invariants of the zero set). –  S. Carnahan Jun 18 '10 at 17:58
    
It makes a big difference whether the coefficients are arbitrary approximately known reals, or simple constants that are known exactly. If you give more detail on this, and the type of polynomials, this might possibly turn into an MO question that gets modded up. –  Dan Piponi Jun 18 '10 at 20:22
    
Many thanks for your kind answers. I now understand that the question is too broad, and its solutions are far beyond my skills. I will work at narrowing my question, and come back here if I cannot find a solution by myself. Tnaks again, Olivier –  Olivier Jun 20 '10 at 15:24
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2 Answers

To add a little to the previous answer, the zeros of a polynomial in more than one variable will not typically form a discrete set: consider the example of $f(x,y)=y-x^2$. The points in the plane which satisfy the equation $f(x,y)=0$ describe a parabola. Your request for an 'algorithm' is too broad to answer here, but many mathematical software packages will plot the zeros for you. In Mathematica, try ContourPlot[f[x,y]==0., ...

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There will be infinitely many. You need as many polynomials as variables to have a finite number. (There is much, much more to say.)

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