For more information about this meeting, contact Mary Anne Raymond.
| Title: | Generating Functions of Rational Polyhedra and Dedekind-Carlitz Polynomials |
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| Seminar: | Combinatorics/Partitions Seminar |
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| Speaker: | Dr. Matthias Beck, San Francisco State University |
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| Abstract: |
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| We study higher-dimensional analogs of the Dedekind-Carlitz polynomials,
c(u,v;a,b) := Sum_{k=1, ..., a-1} u^{k-1} v^{floor(kb/a)} ,
where u and v are indeterminates and a and b are positive integers. These
polynomials satisfy the reciprocity law
(u-1) c(u,v;a,b) + (v-1) c(v,u;b,a) = u^{a-1} v^{b-1} - 1 ,
from which one easily deduces many classical reciprocity theorems for the
Dedekind sum and its generalizations, most notably by Hardy and
Berndt-Dieter.
Dedekind-Carlitz polynomials appear naturally in generating functions of
rational cones. We use this fact to give geometric proofs of the Carlitz
reciprocity law. Our approach gives rise to new reciprocity theorems and a
multivariate generalization of the Mordell-Pommersheim theorem on the
appearance of Dedekind sums in Ehrhart polynomials of 3-dimensional lattice
polytopes.
I will not assume familiarity with Dedekind sums or discrete geometry and I
will carefully define all the terminology used above. The talk will be
accessible to a beginning graduate student.
This is joint work with Asia Matthews (Queens University). |
Room Reservation Information
| Room Number: | MB106 |
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| Date: | 11 / 06 / 2007 |
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| Time: | 11:15am - 12:05pm |
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