Real radical

Summary

In algebra, the real radical of an ideal I in a polynomial ring with real coefficients is the largest ideal containing I with the same vanishing locus. It plays a similar role in real algebraic geometry that the radical of an ideal plays in algebraic geometry over an algebraically closed field. More specifically, the Nullstellensatz says that when I is an ideal in a polynomial ring with coefficients coming from an algebraically closed field, the radical of I is the set of polynomials vanishing on the vanishing locus of I. In real algebraic geometry, the Nullstellensatz fails as the real numbers are not algebraically closed. However, one can recover a similar theorem, the real Nullstellensatz, by using the real radical in place of the (ordinary) radical.

DefinitionEdit

The real radical of an ideal I in a polynomial ring   over the real numbers, denoted by  , is defined as

 

The Positivstellensatz then implies that   is the set of all polynomials that vanish on the real variety[Note 1] defined by the vanishing of  .

ReferencesEdit

  • Marshall, Murray Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. ISBN 978-0-8218-4402-1; 0-8218-4402-4

NotesEdit

  1. ^ that is, the set of the points with real coordinates of a variety defined by polynomials with real coefficients