Workshop "Enumeration and bounds in real algebraic geometry"

 April 21-26, 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.



We are currently inviting participants, and list those
who have indicated a strong desire to attend, or who are in 
residence in the parent program at the CIB.

D.J. Bates
S. Basu
E. Brugalle 
P. Buergisser
A. Degtyarev
A. Gabrielov
M. Hampton 
C. Hillar
I. Itenberg 
M. Joswig
S. Kharlamov 
A. Khovanskii
S. Orevkov
J.M. Rojas
B. Shapiro
B. Shiffman
E. Shustin
E. Soprunova 
F. Sottile
S. Zrebic