with a rational slope passing through the rational point. The complex variety has no -rational points, but has many complex points. If is an affine variety in defined over the complex numbers , the -rational points of can be drawn on a piece of paper or by graphing software. The figure on the right shows the -rational points of Singular points and tangent space Let be an affine variety defined by the polynomials and be a point of . The Jacobian matrix of at is the matrix of the partial derivatives The point is regular if the rank of equals the codimension of , and singular otherwise. If is regular, the tangent space to at is the affine subspace of defined by the linear equations If the point is singular, the affine subspace defined by these equations is also called a tangent space by some authors, while other authors say that there is no tangent space at a singular point. A more intrinsic definition, which does not use coordinates is given by Zariski tangent space. The Zariski topology The affine algebraic sets of kn form the closed sets of a topology on kn, called the Zariski topology. This follows from the fact that and (in fact, a countable intersection of affine algebraic sets is an affine algebraic set). The Zariski topology can also be described by way of basic open sets, where Zariski-open sets are countable unions of sets of the form for These basic open sets are the complements in kn of the closed sets zero loci of a single polynomial. If k is Noetherian (for instance, if k is a field or a principal ideal domain), then every ideal of k is finitely-generated, so every open set is a finite union of basic open sets. If V is an affine subvariety of kn the Zariski topology on V is simply the subspace topology inherited from the Zariski topology on kn. Geometry–algebra correspondence The geometric structure of an affine variety is linked in a deep way to the algebraic structure of its coordinate ring. Let I and J be ideals of k[V], the coordinate ring of an affine variety V. Let I(V) be the set of all polynomials in which vanish on V, and let denote the radical of the ideal I, the set of polynomials f for which some power of f is in I. The reason that