## Jonas Jankauskas, Pure Mathematics, University of Waterloo

### "There are no two non-real conjugates of a Pisot number with the same imaginary part"

In
this
talk,
we
will
present
the
results
from
the
recent
arXiv
paper
by
A.
Dubickas,
K.
G.
Hare
and
J.
Jankauskas
on
the
solution
of
three
and
four
term
linear
equations
in
the
conjugates
of
a
Pisot
number.

More
precisely,
we
show
that
the
number
a=(1+\sqrt{3+2\sqrt{5}})/2
with
minimal
polynomial
x^4-2x^3+x-1
is
the
only
Pisot
number
whose
four
distinct
conjugates
a_1,
a_2,
a_3,
a_4
satisfy
the
additive
relation
a_1+a_2=a_3+a_4.
This
answers
the
earlier
conjecture
due
to
C.
J.
Smyth
and
the
first
author,
namely,
that
there
exists
no
two
non-real
conjugates
of
a
Pisot
number
with
the
same
imaginary
part
and
also
that
at
most
two
conjugates
of
a
Pisot
number
can
have
the
same
real
part.
On
the
other
hand,
we
prove
that
similar
four
term
equations
a_1=a_2+a_3+a_4
or
a_1+a_2+a_3+a_4
=0
cannot
be
solved
in
conjugates
of
a
Pisot
number
a.

We
also
show
that
the
roots
of
the
Siegel's
polynomial
x^3-x-1
are
the
only
solutions
to
the
three
term
equation
a_1+a_2+a_3=0
in
conjugates
of
a
Pisot
number.
Finally,
we
prove
that
there
exists
no
Pisot
number
whose
conjugates
satisfy
the
relation
a_1=a_2+a_3.