  For every compact almost complex manifold (M,J) equipped with a J-preserving
circle action with isolated fixed points, a simple algebraic identity involving
the first Chern class is derived. This enables us to construct an algorithm to
obtain linear relations among the isotropy weights at the fixed points. Suppose
that M is symplectic and the action is Hamiltonian. If the manifold satisfies
an extra "positivity condition" this algorithm determines a family of vector
spaces which contain the admissible lattices of weights.
  When the number of fixed points is minimal, this positivity condition is
necessarily satisfied whenever dim(M)< 8, and, when dim(M)=8, whenever the
S^1-action extends to an effective Hamiltonian T^2-action, or none of the
isotropy weights is 1. Moreover there are no known examples with a minimal
number of fixed points contradicting this condition, and their existence is
related to interesting questions regarding fake projective spaces [Y]. We run
the algorithm for dim(M)< 10, quickly obtaining all the possible families of
isotropy weights. In particular, we simplify the proofs of Ahara and Tolman for
dim(M)=6 [Ah,T1] and, when dim(M)=8, we prove that the equivariant cohomology
ring, Chern classes and isotropy weights agree with the ones of C P^4 with the
standard S^1-action (thus proving the symplectic Petrie conjecture [T1] in this
setting).
