Real radical

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.

Definition

The real radical of an ideal I in a polynomial ring $\mathbb {R} [x_{1},\dots ,x_{n}]$ over the real numbers, denoted by ${\sqrt[{\mathbb {R} }]{I}}$ , is defined as

{\sqrt[{\mathbb {R} }]{I}}=\left\{f\in \mathbb {R} [x_{1},\dots ,x_{n}]\mid -f^{2m}=\sum _{i}h_{i}^{2}+g{\text{ where }}\ m\in \mathbb {Z} _{+},h_{i}\in \mathbb {R} [x_{1},\dots ,x_{n}],{\text{ and }}g\in I\right\}.

The Positivstellensatz then implies that ${\sqrt[{\mathbb {R} }]{I}}$ is the set of all polynomials that vanish on the real variety defined by the vanishing of $I$ .

References

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.