On XOR Lemma for Polynomial Threshold Weight and Length

Let f : {-1, 1}(n) ->{-1, 1} be a Boolean function. We say that a multilinear polynomial p sign-represents f if f(x) = sgn(p(x)) for all x is an element of {-1, 1}(n). In this paper, we consider the length and weight of polynomials sign-representing Boolean functions of the form f circle plus f circle plus ... circle plus f where each f is on a disjoint set of variables. Obviously, if p sign-represents f, then p(x)p(y) sign-represents f(x) circle plus f(y). We give a constructive proof that there is a shorter polynomial when f is AND on n variables for every n >= 3. In addition, we introduce a parameter v(f)(*) of a Boolean function and show that the k-th root of the minimum weight of a polynomial sign-representing f circle plus f circle plus ... circle plus f (k times) converges between v(f)(*) and (v(f)(*))(2) as k goes to infinity.