of points (x, x2, x3) for x in C is an algebraic variety, and more precisely an algebraic curve that is not contained in any plane. It is the twisted cubic shown in the above figure. It may be defined by the equations The irreducibility of this algebraic set needs a proof. One approach in this case is to check that the projection (x, y, z) → (x, y) is injective on the set of the solutions and that its image is an irreducible plane curve. For more difficult examples, a similar proof may always be given, but may imply a difficult computation: first a Gröbner basis computation to compute the dimension, followed by a random linear change of variables (not always needed); then a Gröbner basis computation for another monomial ordering to compute the projection and to prove that it is generically injective and that its image is a hypersurface, and finally a polynomial factorization to prove the irreducibility of the image. General linear group The set of n-by-n matrices over the base field k can be identified with the affine n2-space with coordinates such that is the (i, j)-th entry of the matrix . The determinant is then a polynomial in and thus defines the hypersurface in . The complement of is then an open subset of that consists of all the invertible n-by-n matrices, the general linear group . It is an affine variety, since, in general, the complement of a hypersurface in an affine variety is affine. Explicitly, consider where the affine line is given coordinate t. Then amounts to the zero-locus in of the polynomial in : i.e., the set of matrices A such that has a solution. This is best seen algebraically: the coordinate ring of is the localization , which can be identified with . The multiplicative group k* of the base field k is the same as and thus is an affine variety. A finite product of it is an algebraic torus, which is again an affine variety. A general linear group is an example of a linear algebraic group, an affine variety that has a structure of a group in such a way the group operations are morphism of varieties. Characteristic variety Let A be a not-necessarily-commutative algebra over a field k. Even if A is not commutative, it can still happen that A has a -filtration so that the associated