10.6 Roots of multivariate polynomials

Sage (using the interface to Singular) can solve multivariate polynomial equations in some situations (they assume that the solutions form a zero-dimensional variety) using Gröbner bases. Here is a simple example:

sage: R = PolynomialRing(QQ, 2, 'ab', order='lp')
sage: a,b = R.gens()
sage: I = (a^2-b^2-3, a-2*b)*R
sage: B = I.groebner_basis(); B
[b^2 - 1, a - 2*b]
So $ b=\pm 1$ and $ a=2b$ .

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