Fewnomial bounds from Gale dual systems
Frank Sottile
Texas A&M University

 In 1980, Askold Khovanskii established his fewnomial bound 
 for the number of real solutions to a system of polynomials, 
 showing that the complexity of the set of real solutions to 
 a system of polynomials depends upon the number of monomials 
 and not on the degree.  This fundamental finiteness result 
 in real algebraic geometry is believed to be unrealistically large.

 Recent work with Bihan has led to a new and substantially lower 
 fewnomial bound which is asymptotically optimal.  The bound is 
 obtained by reducing the original system to a Gale dual system,
 and then bounding the number of solutions to a Gale system.  
 
 In this talk, I will describe the history of this problem and
 outline our new results, as well as some applications of this
 new bound.