Bounds on reality

Frank Sottile

In applications of algebraic geometry, it is the real (often positive) solutions
which matter.   The gold  standard for bounds  on the number  of real  zeroes is
Descartes'  bound  for  univariate  polynomials.   Multivariate  generalisations
remain  elusive.  Khovanskii  famously  gave astronomical  fewnomial bounds  for
positive  solutions to  a  system of  equations  that depend  on  the number  of
monomials.   While improved  by  work of  Bihan and  others,  the bounds  remain
unrealistically large.   Bihan's method,  Gale duality ofor  polynomial systems,
replaces a  system of  n polynomials  in n  variables and  n+1+k monomials  by a
system  of   rational  functions  in   a  k-dimensional  polyhedron.    This  is
particularly efficacious when  k=1 (a circuit) where it  gives a Descartes'-like
bound.  I will survey these developments  and lay out some challenges, including
extending this to non-standard real structures.