# 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

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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.

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