Meszaros Tamas:
Some extensions of Alon's Nullstellensatz
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Abstract:
Alon's combinatorial Nullstellensatz, and in particular the 
resulting
nonvanishing criterion is one of the most powerful algebraic
tools in combinatorics, with many important applications. In 
this
talk we extend the
nonvanishing theorem in two directions. We prove a version 
allowing
multiple points.
Also, we establish a variant which is valid over arbitrary
commutative rings, not merely
over subrings of fields. As an application, we prove extensions 
of
the theorem of Alon and
Fueredi on hyperplane coverings of discrete cubes.
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This is joint work with Geza Kos, Tamas Meszaros, Lajos Ronyai.