  We show the existence of systems of n polynomial equations in n variables,
with a total of n+k+1 distinct monomial terms, possessing [n/k+1]^k
nondegenerate positive solutions. (Here, [x] is the integer part of a positive
number x.) This shows that the recent upper bound of
  (e^2+3)/4 2^{\binom{k}{2}} n^k for the number of nondegenerate positive
solutions is asymptotically sharp for fixed k and large n.
  We also adapt a method of Perrucci to show that there are fewer than
  (e^2+3)/4 2^{\binom{k}{2}} 2^n n^k connected components in a smooth
hypersurface in the positive orthant of R^n defined by a polynomial with n+k+1
monomials. Our results hold for polynomials with real exponents.
