Non-linear multivariable functions

[m~ray]

I wanted to know if there is any way of classifying the set of all non-linear multivariable functions. I wish to analyse something over all possible non linear functions with 4 variables. In fact these variables are binary variables. for example f(x,y,u,v)= x.y - u$\oplus$v

 

[fzero]

I wanted to know if there is any way of classifying the set of all non-linear multivariable functions. I wish to analyse something over all possible non linear functions with 4 variables. In fact these variables are binary variables. for example f(x,y,u,v)= x.y - u$\oplus$v

If you are concerned with classification, it is important to explain exactly what structure we have here. Are $x,y,u,v$ elements of some vector spaces? Are the products maps to the same or a different vector space, or perhaps field?

There are some well-studied objects with a structure that might be relevant for your problem. A bilinear map is a function

$$B: V \times W \rightarrow X,$$

where $V, W, X$ are vector spaces and $v \mapsto B(v,w)$, $w \mapsto B(v,w)$ are linear maps.

In the case where $W=V$ and $X = F$ is actually a field, we have a bilinear form:

$$ V : V\times V \rightarrow F.$$

Closely related to this is the notion of a sesquilinear form.

If your function of interest fits one of these categories, it might be possible to find additional information that might be relevant to your question.