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 [math]\displaystyle{ \mathbb{R}[x_1,\dots,x_n] }[/math] over the real numbers, denoted by [math]\displaystyle{ \sqrt[\mathbb{R}]{I} }[/math], is defined as
- [math]\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\}. }[/math]
The Positivstellensatz then implies that [math]\displaystyle{ \sqrt[\mathbb{R}]{I} }[/math] is the set of all polynomials that vanish on the real variety[Note 1] defined by the vanishing of [math]\displaystyle{ I }[/math].
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
Notes
- ↑ that is, the set of the points with real coordinates of a variety defined by polynomials with real coefficients
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