The nullstellensatz says that a system $S$ of polynomial equations $f_1=0,f_2=0,…f_n=0$, where $f_i$ are elements of a polynomial ring over a field $K[x_0,x_1,…]$, will have a non-empty affine variety iff every polynomial in the sistem belongs to a proper ideal $I\subset K[x_0,x_1,…]$. The negation of the theorem says that if $S$ has no solution (and thus, $V(I)=\emptyset$), then $I$ is not a proper ideal, or putting it as Wikipedia states: “there exists $p_0,p_1,…,p_n$ such that $p_0f_0+p_1f_1+…+p_nf_n=1$“. So far i understood the theorem, but I can’t seem to find a sistem $S$ with $V(I)=\emptyset$. Is there any example of a non-trivial $S$ with empty affine variety that you know of?
Edit: Maybe i can’t find any because i presume the elements of $K[x_0,x_1,…]$ can have a constant/independent term (example, $x+y-3$). If it is the case, let me know in the answer.
The most direct example is $V(1)$. All other examples are essentially this example, but dressed up in a more interesting way - $V(f_1,\cdots,f_n)=\emptyset$ implies $(f_1,\cdots,f_n)=(1)$, as the Wikipedia page you quote says.
Here's a bunch of other examples: