Real radical explained

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

R [x _{1,...,x} _{n]} over the real numbers, denoted by \sqrt[ R ]{I} , is defined as\sqrt[ R ]{I} = \ { f \in R [x _{1,...,x} _{n]} \left| - f ^{2m} = style{\sum _{i}} h ^{2} _{i} + g \right. where m \in Z _{+,} h _{i} \in R [x _{1,...,x} _{n],} and g \in I \ }.

The Positivstellensatz then implies that

\sqrt[ R ]{I} is the set of all polynomials that vanish on the real variety^[1] 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. ; 0-8218-4402-4

Notes and References

1. that is, the set of the points with real coordinates of a variety defined by polynomials with real coefficients