On computing Boolean functions by sparse real polynomials

We investigate the complexity of Boolean functions f with respect to realizations by real polynomials p (voting polynomials) in the sense that the sign of p(x) determines the value f(x). Considerable research has been done on determining the minimal degree needed for realizing or approximating particular functions. In this paper we focus our interest on estimating the minimal number of monomials, i.e. the length of realizing polynomials. Our main observation is that, in contrast to the degree, the minimal length essentially depends on whether we realize f over the domain {0,1}ⁿ (which corresponds to threshold-and circuits for f), or over the domain {1,-1}ⁿ (which corresponds to threshold-parity circuits)