Sharpness of fewnomial bounds and the number of components of a fewnomial hypersurface

We give a construction of a nxn fewnomial system with n+k+1 monomials having k^-k(n+k)^k positive solutions. This shows that the dependence on n in the fewnomial upper bound of (e²+3) 2^k(k-1)/2-2n^k is sharp, for k fixed. We also adapt a method of Perrucci to show that there are fewer than (e²+3) 2^k(k-1)/2-22ⁿn^k connected components in a hypersurface in the positive orthant of Rⁿ defined by a polynomial with n+k+1 monomnials. Our results hold for polynomials with real exponents.