  We show that detecting real roots for honestly n-variate (n+2)-nomials (with
integer exponents and coefficients) can be done in time polynomial in the
sparse encoding for any fixed n. The best previous complexity bounds were
exponential in the sparse encoding, even for n fixed. We then give a
characterization of those functions k(n) such that the complexity of detecting
real roots for n-variate (n+k(n))-nomials transitions from P to NP-hardness as
n tends to infinity. Our proofs follow in large part from a new complexity
threshold for deciding the vanishing of A-discriminants of n-variate
(n+k(n))-nomials. Diophantine approximation, through linear forms in
logarithms, also arises as a key tool.
