Encyclopedia > Dimension of an algebraic variety

  Article Content

Dimension of an algebraic variety

In algebraic geometry, the dimension of an algebraic variety V is defined, informally speaking, as the number of independent rational functions that exist on V. So, for example, an algebraic curve has by definition dimension 1. That means that any two rational functions F and G on it satisfy some polynomial relation P(F,G) = 0. That implies that F and G are constrained to take related values (up to some finite freedom of choice): they cannot be truly independent.

Formal definition

For an algebraic variety V over a field K, the dimension of V is the transcendence degree[?] over K of the function field[?] K(V) of all rational functions[?] on V, with values in K.

For the function field even to be defined, V here must be an irreducible algebraic set; in which case the function field (for an affine variety) is just the field of fractions of the co-ordinate ring of V. It is easy to define by polynomials sets that have 'mixed dimension': a union of a curve and a plane in space, for example. These fail to be irreducible.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
58 BC

... Contents 58 BC Centuries: 2nd century BC - 1st century BC - 1st century Decades: 100s BC 90s BC 80s BC 70s BC 60s BC - 50s BC - 40s BC 30s BC 20s BC 10s BC 0s ...