On the Sharpness of fewnomial bound and the number of components of a fewnomial hypersurface

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.