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 (real) 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, Hilbert's 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 ${\displaystyle \mathbb {R} [x_{1},\dots ,x_{n}]}$ over the real numbers, denoted by ${\displaystyle {\sqrt[{\mathbb {R} }]{I}}}$, is defined as

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

The Positivstellensatz then implies that ${\displaystyle {\sqrt[{\mathbb {R} }]{I}}}$ is the set of all polynomials that vanish on the real variety^{[Note 1]} defined by the vanishing of ${\displaystyle I}$.

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

Notes