I am aware of companion matrices for single variable polynomials:
$$p(x) = x^k+c_{k-1}x^{k-1}+\cdots +c_0$$
$${\bf C_p} = \left[\begin{array}{ll}{\bf 0}^T & -c_0 \\ {\bf I}_{k-1} & {\bf c}_{1:k}\end{array}\right]$$
$\bf C_p$ will have the same eigenvalues as the roots of $p(x)=0$
Are there any ways to build companion matrices for multivariate polynomials (in some sense)?
EDIT for example what I would like is that if I have $f(x,y) = (x+1)^2+(y+1)^2-1$ I can (by some recipe or algorithm) build a matrix that has the same properties in the sense of approximating roots of the polynomial when iterating matrix multiplication in the same way as one variable do.