  Polynomial threshold gates are basic processing units of an artificial neural
network. When the input vectors are binary vectors, these gates correspond to
Boolean functions and can be analyzed via their polynomial representations. In
practical applications, it is desirable to find a polynomial representation
with the smallest number of terms possible, in order to use the least possible
number of input lines to the unit under consideration. For this purpose,
instead of an exact polynomial representation, usually the sign representation
of a Boolean function is considered. The non-uniqueness of the sign
representation allows the possibility for using a smaller number of monomials
by solving a minimization problem. This minimization problem is combinatorial
in nature, and so far the best known deterministic algorithm claims the use of
at most $0.75\times 2^n$ of the $2^n$ total possible monomials. In this paper,
the basic methods of representing a Boolean function by polynomials are
examined, and an alternative approach to this problem is proposed. It is shown
that it is possible to use at most $0.5\times 2^n = 2^{n-1}$ monomials based on
the $\{0, 1\}$ binary inputs by introducing extra variables, and at the same
time keeping the degree upper bound at $n$. An algorithm for further reduction
of the number of terms that used in a polynomial representation is provided.
Examples show that in certain applications, the improvement achieved by the
proposed method over the existing methods is significant.
