Computational Algebra Seminar: Garzella -- Zeta functions on projective hypersurfaces via controlled reduction

Speaker: Jack J. Garzella  
Title: Zeta functions on projective hypersurfaces via controlled reduction   
Time & Place: 15.00-16.00, Thursday 23 April, SMRI Seminar Rm
Abstract: The zeta function of a variety in characteristic p captures a lot of
arithmetic information about that variety.  Calculating this zeta function as fast as
possible is a classical problem in computational number theory.  We describe a
cohomological approach called *controlled reduction*, due to Costa and Harvey, which is
the state of the art for many varieties of dimension greater than one.  We describe
various ways one can improve the algorithms of Costa and Harvey, including an "abstract
controlled reduction problem" which abstracts the algorithm away from the specifics of
any particular class of varieties.  Using our algorithms, we find many examples of
varieties with interesting arithmetic invariants (like Newton polygons and domino
numbers).  All work is joint with Batubara, Huang, and Mellberg.