Posted in Algebraic Geometry and Analytic Geometry, Learning

Nullstellensatz in the real case

In alegebraic geometry, we have Hilbert Nullstellensatz and in real algebraic geometry we have Real Nullstelllensatz. There is a difference between these theorems.

Strong Hilbert Nullstellensatz: I(V_{\mathbb{C}}(I)) = \sqrt{I}.

If polynomial f is vanish on the set \begin{cases}f_1 &= 0 \\ &\dots \\ f_k &= 0\end{cases} (in \mathbb{C}^n ) then f has  the following form:

f \in\sqrt{I} , that is: \exists m \in \mathbb{N}: f^m = g_1f_1 + \dots + g_kf_k với g_j \in \mathbb{R}[X_1, \dots,X_n].

In the case of \mathbb{R}^n

Real Nullstellensatz: I(V_{\mathbb{R}}(I)) = \sqrt[\mathbb{R}]{I}.

V_{\mathbb{C}}(I) \cap \mathbb{R}^n = V_{\mathbb{R}}(I):

f^{2s} \in -(\sum \mathbb{R}[X]^2 + I)  or f^{2s} + \sum_{j =1}^m p_j^2 = h_1f_1 + \dots + h_kf_k .