Example of a boolean function Is there an example of real polynomial representation of a Boolean function with $4$ variables whose polynomial degree is $2$ that depends on $4$ variables?
Yes. Using the convention that inputs and outputs are $\pm 1$, then the function is
$$
\frac{a(x+y)+b(x-y)}{2}.
$$
You could have found this function using exhaustive search – there are only $2^{16}$ functions on $4$ variables.