---
tag: 4M94FZH9Y8
kind: concept
name: Regular function
msc: [14A10]
sources:
  - url: https://en.wikipedia.org/wiki/Morphism_of_algebraic_varieties
---

## 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.

## Explanations

### L1 human

On the affine line, a regular function on the whole line is just a polynomial.
On the line with the origin removed, the function $1/x$ is also regular, because its denominator never vanishes there.
In general a function is regular when near each point it can be written as a fraction of polynomials whose bottom part is nonzero at that point.
On a projective variety, the only regular functions on the whole variety are the constants.