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 formula_1 over the real numbers, denoted by formula_2, is defined as
The Positivstellensatz then implies that formula_2 is the set of all polynomials that vanish on the real variety defined by the vanishing of formula_5.