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A polynomial in one variable can be expressed as a companion matrix, of which the eigenvalues are the roots of the polynomial and which can be found by using e.g. QR decomposition or power iteration.

Is there anything like this for multivariate polynomials? To keep things simple: is there a "companion" matrix for quadratic or cubic bivariate polynomial? And is it possible to employ some eigenvalue algorithm for find all of its roots?

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I think that this paper would be a great place to start.

I'd also recommend this, which constructs a generalization of companion matrices, in this case ring representations.

So yes, there is a sort of analog in which an eigenvalue algorithm is used.

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In general you can't find all of multivariate polynomial roots. You can factor it, but the factorization is in general not unique and you won't be getting just one root per factor. For example $$x^2+y^2 - 1=0$$ has solutions all along the unit circle so that would be an infinite number of roots. But if you have $$x^3+xy^2-x =0$$ of course you can write it as $$x(x^2+y^2-1)=0$$ which is the selection between a line of $0$ for an $x$ and said circle.

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