Characteristic polynomials of Boolean functions

In this article we introduce the notion of a characteristic polynomial of a Boolean function having a given polarization of variables and consider a method for representing a Boolean function by the Reed–Muller polynomial (the canonical polarized polynomial) using the characteristic polynomial of this function.
It is proved that the values of the characteristic polynomial coincide with the corresponding coefficients of the Reed–Muller polynomial and a linear algorithm for finding the coefficients of the Reed–Muller polynomial is presented.
We also consider positively polarized characteristic polynomials and problems associated with them, including checking whether the Boolean function belongs to the class of linear functions.
Examples of the application of characteristic polynomials to the determination of Reed–Muller polynomials, the extension of a partial Boolean function to a linear function and the verification of a Boolean function for linearity are given.