INVESTIGADORES
DICKENSTEIN Alicia Marcela
congresos y reuniones científicas
Título:
Counting solutions to binomial systems
Autor/es:
A. DICKENSTEIN
Lugar:
Stuttgart, Alemania
Reunión:
Conferencia; Colloquium; 2006
Resumen:
Binomial ideals are quite ubiquitous in very different contexts
particularly those involving toric geometry and its applications
in the study of semigroup algebras, and in the modern versions
of hypergeometric systems of differential equations.
In this talk we consider ideals generated by n binomials in a
polynomial ring in n variables with coefficients in a field k.
We are interested in determining when the number of affine solutions
over the algebraic closure of k is finite and non zero (i.e. when the
given binomials define a complete intersection) and, in this case,
to count the number of solutions, with or without multiplicity,
directly in terms of the given data: the 2n exponents and the
2n coefficients.
We use commutative algebra tools to reduce the study of these
solutions to a combinatorial problem on a graph associated to the
exponents occurring in the given binomials. It follows that for
generic exponents the problem of counting the number of solutions
(with or without multiplicity) can be solved in polynomial time,
but that the general problem is #P-complete, i.e. at least as hard as
an NP-complete problem.
If time permits, we will outline some applications to differential
equations and to the computation of sparse discriminants. This is
joint work with Eduardo Cattani.