the equations and inequations are all the points of an algebraic curve in which a finite number of points have been removed. This curve is defined by a system of generators of the ideal of the polynomials h such that it exists an integer k such belongs to the ideal generated by . This representation is a birational equivalence between the curve and the plane curve defined by f. Every algebraic curve may be represented in this way. However, a linear change of variables may be needed in order to make almost always injective the projection on the two first variables. When a change of variables is needed, almost every change is convenient, as soon as it is defined over an infinite field. This representation allows us to deduce easily any property of a non-plane algebraic curve, including its graphical representation, from the corresponding property of its plane projection. For a curve defined by its implicit equations, above representation of the curve may easily deduced from a Gröbner basis for a block ordering such that the block of the smaller variables is (x1, x2). The polynomial f is the unique polynomial in the base that depends only of x1 and x2. The fractions gi/g0 are obtained by choosing, for i = 3, ..., n, a polynomial in the basis that is linear in xi and depends only on x1, x2 and xi. If these choices are not possible, this means either that the equations define an algebraic set that is not a variety, or that the variety is not of dimension one, or that one must change of coordinates. The latter case occurs when f exists and is unique, and, for i = 3, …, n, there exist polynomials whose leading monomial depends only on x1, x2 and xi. Algebraic function fields The study of algebraic curves can be reduced to the study of irreducible algebraic curves: those curves that cannot be written as the union of two smaller curves. Up to birational equivalence, the irreducible curves over a field F are categorically equivalent to algebraic function fields in one variable over F. Such an algebraic function field is a field extension K of F that contains an element x which is transcendental over F, and such that K is a finite algebraic extension of F(x), which is the field of rational functions in the indeterminate x over F. For