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Notion of a rational function on an algebraic variety

Like polynomials, rational expressions can also be generalized to n indeterminates X,..., X, by taking the field of fractions of F, which is denoted by F (X,..., X).

An extended version of the abstract idea of rational function is used in algebraic geometry. There the function field of an algebraic variety V is formed as the field of fractions of the coordinate ring of V (more accurately said, of a Zariski - dense affine open set in V). Its elements f are considered as regular functions in the sense of algebraic geometry on non-empty open sets U, and also may be seen as morphisms to the projective line.