This paper is our third step towards developing a theory of testing monomials
in multivariate polynomials and concentrates on two problems: (1) How to
compute the coefficients of multilinear monomials; and (2) how to find a
maximum multilinear monomial when the input is a ΠΣΠ polynomial. We
first prove that the first problem is \#P-hard and then devise a O∗(3ns(n))
upper bound for this problem for any polynomial represented by an arithmetic
circuit of size s(n). Later, this upper bound is improved to O∗(2n) for
ΠΣΠ polynomials. We then design fully polynomial-time randomized
approximation schemes for this problem for ΠΣ polynomials. On the
negative side, we prove that, even for ΠΣΠ polynomials with terms of
degree ≤2, the first problem cannot be approximated at all for any
approximation factor ≥1, nor {\em "weakly approximated"} in a much relaxed
setting, unless P=NP. For the second problem, we first give a polynomial time
λ-approximation algorithm for ΠΣΠ polynomials with terms of
degrees no more a constant λ≥2. On the inapproximability side, we
give a n(1−ϵ)/2 lower bound, for any ϵ>0, on the
approximation factor for ΠΣΠ polynomials. When terms in these
polynomials are constrained to degrees ≤2, we prove a 1.0476 lower
bound, assuming P=NP; and a higher 1.0604 lower bound, assuming the
Unique Games Conjecture