Say I have a set of points in $\mathbb{Z}^3 \times \mathbb{Z}_2$ each of which represent part of a mapping $(z_1, z_2, z_3) \mapsto z_4 \in \mathbb{Z}_2$. How do I find the the simplest polynomial function such that the restriction to the above set of given points $\{(z_1,z_2,z_3)\}$ is equal to the given mapping. I.e. for all $(z_1,z_2,z_3,z_4)\in \mathbb{Z}^3\times \mathbb{Z}_2$, $ \ f(z_1,z_2,z_3) = z_4$?
Please see a related post here.
I think you can take the set of points and modulo the coordinates by $2$ since the polynomials will be evaluated mod $2$ and taking an integer modulo $2$ is a homomorphism and distributes through the polynomial using the homomorphism. Then you have a table of inputs and outputs:
$$ \begin{vmatrix} (z_1, & z_2, & z_3)& \rightarrow & z_4 \\ \hline 0 & 1 & 1 & \rightarrow &0 \\ 1 & 0 & 1 & \rightarrow &1 \\ {\vdots} \\ 1 & 1 & 1 & \rightarrow &1 \\ 0 & 1 & 0 & \rightarrow &1 \end{vmatrix} $$
of mappings that is finite. Somehow you then turn this into a Boolean function in normal form, but forgot how.