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 $$\mathbb{R}[x_1,\dots,x_n]$$ over the real numbers, denoted by $$\sqrt[\mathbb{R}]{I}$$, is defined as
 * $$\sqrt[\mathbb{R}]{I} = \Big\{ f \in \mathbb{R}[x_1,\dots,x_n] \left|\, -f^{2m} = \textstyle{\sum_i} h_i^2 + g \right.\text{ where }\ m \in \mathbb{Z}_+,\, h_i \in \mathbb{R}[x_1,\dots,x_n], \,\text{and } g \in I\Big\}.$$

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