how do you find the highest common factor of two multivariate polynomials?

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How do you find the highest common factor of two multivariate polynomials?

I am happy to get answers that are only useful for polynomials over the real numbers, as that is what I am dealing with.

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This is discussed in Cox, Little, and O'Shea's book "Ideals, Varieties, and Algorithms" around page 180. There is also the command PolynomialGCD in mathematica, and it appears to work on wolfram alpha.