The number of solutions of a polynomial system using a Gröbner basis

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Yes, these

monomials that are not a multiple of any leading monomial of G

form a basis of the quotient ring $R/I$ (by the reduction algorithm), also called a normal basis for $I$.

sage: R.<x,y,z> = PolynomialRing(QQ,3)
sage: I = Ideal([x^2+y+z-1,x+y^2+z-1,x+y+z^2-1])
sage: B = I.groebner_basis(); B
[x^2 + y + z - 1, y^2 + x + z - 1, z^2 + x + y - 1]
sage: I.normal_basis()
[x*y*z, y*z, x*z, z, x*y, y, x, 1]

The number of elements in this basis, or the vector space dimension of $R/I$, can also be found as follows:

sage: I.vector_space_dimension()
8

The zero set, or variety, of the ideal is the following:

sage: I.variety(QQbar)
[{z: 0, y: 0, x: 1},
 {z: 0, y: 1, x: 0},
 {z: 1, y: 0, x: 0},
 {z: -2.414213562373095?, y: -2.414213562373095?, x: -2.414213562373095?},
 {z: 0.4142135623730951?, y: 0.4142135623730951?, x: 0.4142135623730951?}]

The number of elements is not 8, because we are missing the "multiplicities". Indeed,

 sage: I.radical() == I
 False

So perhaps the more interesting quantity is

 sage: I.radical().vector_space_dimension()
 5

which agrees with the number of zeros without multiplicity.

(There is an easy algorithm to compute the radical of a zero-dimensional ideal.)