- Source: 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
R
[
x
1
,
…
,
x
n
]
{\displaystyle \mathbb {R} [x_{1},\dots ,x_{n}]}
over the real numbers, denoted by
I
R
{\displaystyle {\sqrt[{\mathbb {R} }]{I}}}
, is defined as
I
R
=
{
f
∈
R
[
x
1
,
…
,
x
n
]
|
−
f
2
m
=
∑
i
h
i
2
+
g
where
m
∈
Z
+
,
h
i
∈
R
[
x
1
,
…
,
x
n
]
,
and
g
∈
I
}
.
{\displaystyle {\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
I
R
{\displaystyle {\sqrt[{\mathbb {R} }]{I}}}
is the set of all polynomials that vanish on the real variety defined by the vanishing of
I
{\displaystyle I}
.
References
Marshall, Murray Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. ISBN 978-0-8218-4402-1; 0-8218-4402-4
Notes
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