Regular function

Summary

A regular function on a variety is a function to the base field that is locally a quotient of two polynomials. The denominator must not vanish where the quotient is used. Regular functions are the algebraic analogue of smooth or holomorphic functions. On an affine variety the regular functions are exactly the polynomial functions, so they form the coordinate ring. They are used to define morphisms of varieties and the structure sheaf of a variety.