Workshop "Enumeration and bounds in real algebraic geometry"

 April 21-25, 2008

   Calculation and estimation of  the number of real (and sometimes positive)  solutions of a system
of  real algebraic  equations is  a question  of great  theoretical and  practical  importance.  The
classical Bezout theorem often provides unsatisfactory bounds, and particular applications typically
require very specific  tools to establish realistic bounds.  For example,  the problem of finiteness
of central configurations in  celestial mechanics was considered by Euler and  still an active field
of  research.  (Recently Moeckel  and his  collaborators bounded  the number  of equilibira  for the
four-body problem).  A  related classical question concerning the number of  (real) roots of abelian
integrals is of enormous value in the infinitesimal version of Hilbert's 16-th problem.  Recent work
of Yakovenko-Novikov represents an important advance on this question.

    It may be more practical to understand  the average numer of real solutions.  That is, determine
the  expected number  of  solutions to  systems of  equations  selected from  a natural  probability
distribution.  There is a long history of  work in this area, and Buergisser recently found formulas
which unify many  known results and enable  the computation of the expected  Euler characteristic of
manifolds  defined by equations  selected from  certain probability  distributions.  Recent  work by
Zelditch-Shiffman and Shiffman's  student Zrebiec involves the related  question of the distribution
of zeroes of random polynomial systems.

   When considering positive solutions, it is  natural to allow polynomials whose exponents are real
numbers  (equivalently, systems  of  exponential functions  on  R^n).  In  his well-known  monograph
'Fewnomials',  (translations   of  Mathematical  Monographs,  88.   American  Mathematical  Society,
Providence, RI, 1991) Khovanskii developed what is now known as fewnomial theory.  In particular, he
gave a famous upper bound for the number of real solutions of a system of n fewnomial equations in n
real  variables,  which was  known  to be  not  sharp.   This was  recently  improved  in papers  of
Bihan-Sottile and coauthors.  There is room for furter improvement in the this new bound, and it has
not yet been widely applied.

   The goal  of this small workshop  is to bring together  these experts and others  working in this
area for  a week  of in-depth discussion  and collaboration.   It is made  possible by  the generous
support of the US National Science Foundation, and the Centre Interfacultaire Bernoulli.

Participants:

 S. Basu        
 D. Bates           
 B. Bertrand 
 F. Bihan    
 E. Brugalle
 P. Buergisser
 A. Degtyarev
 S. Erdogan
 A. Gabrielov    
 M. Hampton    
 I. Itenberg
 M. Joswig 
 S. Kharlamov
 A. Khovanskii 
 M. Nisse
 D. Novikov 
 S. Orevkov
 N. Puignau
 R. Rasdeaconu 
 E. Rey
 M. Rojas       
 I. Scherbak 
 B. Shapiro
 B. Shiffman   
 E. Shustin
 E. Soprunova   
 F. Sottile       
 A. Vainstein
 E. Will
 S. Yakovenko 
 S. Zrebiec