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 .