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
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.
Modified since: 13 December 2007.