Counting solutions of singular mean field equations on tori via algebraic geometry
林耿立博士生國立中央大學 上午 11:00–12:00 鴻經館 107 室It was conjectured by C.-S. Lin and C.-L.Wang that the number Nk of solutions
for the singular mean field equation on tori
Δu + peu = pδ0
with coefficient r = 4π(2k + 1) is equal to its topological degree k + 1 for
each k ? Z≧0.
It turns out that such a non-linear PDE is indeed algebraically integrable
and we may reduce the problem to a related one on counting solutions of
certain affine polynomial systems.
We use projective B´ezout theorem and residual intersection theory to verify
the conjecture for k ≦ 5. The structure of the redundant infinity solution
for the projectivized system involves some symmetric combinatorics and delicate
polyhedron structures. It seems that these structures are related to
many other fields of mathematics and deserve further study.