Real radical

Template:Short description 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 $ℝ[x_1,\dots ,x_n]$ over the real numbers, denoted by $\sqrt[ℝ]{I}$, is defined as

$\sqrt[ℝ]{I}=\{f\in ℝ[x_1,\dots ,x_n]|-f^{2m}={\sum }_i{h}_i^2+g\text{ where }m\in {\mathbb{Z}}_{+},h_i\in ℝ[x_1,\dots ,x_n],\text{and }g\in I\}.$

The Positivstellensatz then implies that $\sqrt[ℝ]{I}$ 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. Template:ISBN; 0-8218-4402-4

Notes

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