regular function

Regular functions are the 'nice' functions of algebraic geometry — the analogue of smooth functions in differential geometry or holomorphic functions in complex analysis. Near any point they look like an honest ratio of polynomials whose denominator does not vanish, so they have no poles where they are defined.

On an open subset U of a variety, a function f is regular at a point p if there is a neighborhood of p and polynomials g, h with h nonvanishing on that neighborhood such that f equals g/h there. It is regular on U if it is regular at every point. The regular functions on U form a ring, and assigning to each open U its ring of regular functions defines the structure sheaf of the variety.

On an affine variety X the global regular functions are exactly the coordinate ring k[X] — no genuine fractions are needed globally — which is why polynomial functions suffice in the affine world. But globally on an irreducible projective variety the only regular functions are constants, since a nonconstant global one would be an unbounded analogue with poles. This contrast is the reason projective geometry is organized around sheaves and graded rings rather than a single function ring.