  We derive new bounds of fewnomial type for the number of real solutions to
systems of polynomials that have structure intermediate between fewnomials and
generic (dense) polynomials. This uses a modified version of Gale duality for
polynomial systems. We also use stratified Morse theory to bound the total
Betti number of a hypersurface defined by such a dense fewnomial. These bounds
contain and generalize previous bounds for ordinary fewnomials obtained by
Bates, Bertrand, Bihan, and Sottile.

Algebraic Geometry