Posted in Algebraic 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 $f^{2s} + \sum_{j =1}^m p_j^2 = h_1f_1 + \dots + h_kf_k$.