Weak Nullstellensatz

In algebraic geometry, one of  the fundamental theorems is Hilbert’s Nullstellensatz.

Weak version:

Weak Hilbert’s Nullstellensatz: If k is algebraic closed field then the maximal ideals of k[x_1, \dots, x_n] are exactly of the form (x_1 - a_1, \dots, x_n -a_n) for some a_i \in k.

Some cases:

If k = \mathbb{C}, a system of polynomials equation have a root ’cause V(I) \ne \emptyset.

If k = \mathbb{R}, there exists an ideal I \ne (x_1 - a_1, \dots, x_n -a_n) such that V(I) = \emptyset.

Another version:
\begin{theorem} Ideal I \subseteq k[X_1, \dots, X_n] with k is algebraic closed. Then V(I) = \emptyset implies I = k[X_1, \dots,X_n].
\end{theorem}

Moreover, k = \mathbb{C}, if we have a system of polynomials equation (f_1 = 0, f_2 = 0, \dots, f_k = 0),  if this one have no root then there exist g_1, \dots, g_k \in \mathbb{C}[X_1,\dots,X_n]

s.t. f_1(X)g_1(X) + f_2(X)g_2(X) + \dots + f_k(X)g_k(X) = 1.

For example (Parrilo): Consider following polynomials over \mathbb{C}:

f_1(x) = x^2 + y^2 - 1 =0,

f_2(x) = x + y = 0,

f_3(x) = 2x^3 + y^3 +1 = 0.

There exist:

g_1(x) = \frac{1}{7}(1 - 16x - 12y - 18xy - 6y^2),

g_2(x) = \frac{1}{7}(-7y - x +4y^2 - 16 + 12xy + 2y^3 + 6y^2x),

g_3(x) = \frac{1}{7}(8 + 4y)

s.t. f_1g_1 + f_2g_2 + f_3g_3 = 1.

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