Analytic and probabilistic combinatorics for polynomials over finite fields
| Speaker: | Daniel Panario |
|---|---|
| Affiliation: | University of Waterloo |
| Room: | William G. Davis Computer Research Centre (DC) 1302 |
Abstract:
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
for
example
on
probability
and
probabilistic
combinatorics.
We
conclude
providing
several
open
problems
of
polynomials
over
finite
fields
related
to
number
theory.