# 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 $\mathbb {R} [x_{1},\dots ,x_{n}]$ over the real numbers, denoted by ${\sqrt[{\mathbb {R} }]{I}}$ , is defined as

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

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