Let $K$ be an algebraically closed field, and let $I$ be an ideal in $K[x_1,\ldots,x_n]$ the polynomial ring in $n$ indeterminates.

Define $V(I)$ the zero set of $I$ by $$ V(I) = \{(a_1,\ldots,a_n) \in K^n \mid f(a_1,\ldots,a_n)=0 \text{ for all } f \in I\}$$

Weak Nullstellensatz:
If $V(I)=\emptyset$ then $I=K[x_1,\ldots,x_n]$ In other words, the zero set of any proper ideal of $K[x_1,\ldots,x_n]$ is nonempty.

Hilbert's (Strong) Nullstellensatz:
Suppose $f \in K[x_1,\ldots,x_n]$ satisfies $f(a_1,\ldots,a_n)=0$ for every $(a_1,\ldots,a_n) \in V(I)$ Then $f^r \in I$ for some integer $r>0$

In the language of algebraic geometry, the latter result is equivalent to the statement that $\operatorname{Rad}(I)=I(V(I))$ that is, the radical of $I$ is equal to the ideal of $V(I)$