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.
Definition edit
The real radical of an ideal I in a polynomial ring over the real numbers, denoted by , is defined as
![{\displaystyle \mathbb {R} [x_{1},\dots ,x_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1df2ced87a73a310f03896a55f51a097cd9876ac)
![{\displaystyle {\sqrt[{\mathbb {R} }]{I}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f35198ff7ea1557693742867ebe5a8c9af87b117)
![{\displaystyle {\sqrt[{\mathbb {R} }]{I}}={\Big \{}f\in \mathbb {R} [x_{1},\dots ,x_{n}]\left|\,-f^{2m}=\textstyle {\sum _{i}}h_{i}^{2}+g\right.{\text{ where }}\ m\in \mathbb {Z} _{+},\,h_{i}\in \mathbb {R} [x_{1},\dots ,x_{n}],\,{\text{and }}g\in I{\Big \}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb20805a672318f6b2a46dfaa3c7feea286903c)
The Positivstellensatz then implies that is the set of all polynomials that vanish on the real variety[Note 1] defined by the vanishing of .
References edit
- 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
Notes edit
- ^ that is, the set of the points with real coordinates of a variety defined by polynomials with real coefficients