Commutative algebra arising from the ADG conjectures

Winfried Bruns

MSC 2000

05A15 Exact enumeration problems, generating functions

05E99 None of the above, but in this section

13F20 Polynomial rings and ideals; rings of integer-valued polynomials

14M25 Toric varieties, Newton polyhedra

52B20 Lattice polytopes

Abstract
The article describes the application of commutative algebra to the solution of enumerative problems arising from linear diophantine systems. This connection was created by Stanley, starting from his work on the Anand-Dumir-Gupta conjectures for the number of "magic squares". The article is to a large extent self-contained, and should be accesiible also to non-specialists. It is based on the authors lectures at the workshop at the HRI, Allahabad, December 2003.