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   Seminário de Teoria da Computaçăo e Combinatória (TCC)

               Tarde Uruguaia de Combinatória
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Título:      Analytic and Probabilistic Combinatorics for 
             Polynomials over Finite Fields

Palestrante: Daniel Panario
             Carleton University

Hora e Data: 15h30m, quarta-feira, 18 de junho de 2014

Local:       Sala Multi-usos do Numec

Resumo:

The central objects of  this talk are univariate polynomials
over  finite fields.   We  first review  a methodology  from
analytic combinatorics that allows not only the study of the
decomposition  of polynomials  into its  irreducible factors
but also the derivation of algorithmic properties as well as
the estimation of average-case analysis of algorithms.  This
methodology  can  be   naturally  used  to  provide  precise
information  on the  factorization of  polynomials  into its
irreducible factors similar to  the classical problem of the
decomposition  of integers into  primes.  Examples  of these
results  are provided.   The  shape of  a random  univariate
polynomial over a finite field is also given.

Then, we briefly show several results for random polynomials
over   finite  fields   that  were   obtained   using  other
methodologies   based  on   probability   and  probabilistic
combinatorics.   For example we  comment on  a study  of the
joint degree  distribution of  the irreducible factors  of a
polynomial by  means of a discrete random  process.  We also
comment of an Erdös--Turán type of approach for the study of
the  expected degree  of  the splitting  field  of a  random
polynomial over a finite field.

We conclude providing some open problems.