Monday, November 5, 2018 4:00 pm
-
4:00 pm
EST (GMT -05:00)
Michael Coons, University of Newcastle
"Mahler’s methods: theorems, speculations and variations"
Mahler's method in number theory is an area wherein one answers questions surrounding the transcendence and algebraic independence of both power series $F(z)$, which satisfy the functional equation a0(z)F(z)+a1(z)F(zk)+⋯+ad(z)F(zkd)=0
for
some
integers
$k\geqslant
2$
and
$d\geqslant
1$
and
polynomials
$a_0(z),\ldots,a_d(z)$,
and
their
special
values
$F(\alpha)$,
typically
at
algebraic
numbers
$\alpha$. The
most
important
examples
of
Mahler
functions
arise
from
important
sequences
in
theoretical
computer
science
and
dynamical
systems,
and
many
are
related
to
digital
properties
of
sets
of
numbers.
For
example,
the
generating
function
$T(z)$
of
the
Thue-Morse
sequence,
which
is
known
to
be
the
fixed
point
of
a
uniform
morphism
in
computer
science
or
equivalently
a
constant-length
substitution
system
in
dynamics,
is
a
Mahler
function.
In
1930,
Mahler
proved
that
the
numbers
$T(\alpha)$
are
transcendental
for
all
non-zero
algebraic
numbers
$\alpha$
in
the
complex
open
unit
disc. With
digital
computers
and
computation
so
prevalent
in
our
society,
such
results
seem
almost
second
nature
these
days
and
thinking
about
them
is
very
natural.
But
what
is
one
really
trying
to communicate
by
proving
that
functions
or
numbers
such
as
those
considered
in
Mahler’s
method?
In this talk, highlighting work from the very beginning of Mahler’s career, we speculate---and provide some variations---on what Mahler was really trying to understand.
MC 5501