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[Solved]: Example of a boolean function

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Problem Detail:

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.