Towards a multivariate Descartes' Rule

  The gold standard in understanding the real solutions to 
systems of equations is  Descartes' rule of signs (c. 1637),
which bounds the number of positive zeroes of a univariate 
polynomial by the number of changes in the signs of its 
coefficients.  Ignoring signs, a univariate polynomial with 
d+2 terms has at most d+1 positive zeroes.

  For multivariate polynomials, the situation is not settled.
In 1980 Khovanskii gave a version of this second estimate, which
was refined a decade ago, but is considered to be far from 
sharp.  Recently, there have been several results giving 
bounds related to signs of coefficients that may suggest 
aspects of an eventual multivariate Descartes' rule.  My talk
will discuss the history and these recent results.