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Small limit points of sets of algebraic integers
Bibliografi
Author:
Garth, David Ryan
;
Cochrane, Todd
(Advisor)
Topik:
MATHEMATICS
Bahasa:
(EN )
ISBN:
0-599-75857-0
Penerbit:
KANSAS STATE UNIVERSITY
Tahun Terbit:
2000
Jenis:
Theses - Dissertation
Fulltext:
9970809.pdf
(0.0B;
0 download
)
Abstract
An algebraic integer α > 1 is called a Pisot number if all of its conjugates lie in the open unit disk. The set
S
of all the Pisot numbers has many interesting properties and has therefore been a prominent topic of research in the past fifty years. It is natural to try to generalize the set of Pisot numbers by defining sets of algebraic integers with any number of conjugates outside the unit circle. We say that an algebraic integer α is in
S
k
if exactly
k
of its conjugates, including α itself, lie outside the unit circle, while the rest of its conjugates lie inside the unit circle. In this work we are interested in the extent to which some of the well known properties of the Pisot numbers have analogs in the sets
S
k
for
k
> 1. One of the fascinating results concerning the set of Pisot numbers is that its smallest limit point is the golden ratio. Given some of the known similarities between the Pisot numbers and the set
S
′′
2
of non-real complex elements of
S
2
, we conjecture that the smallest modulus of a limit point of
S
′′
2
is the square root of the golden ratio. While we are unsuccessful in our attempt to prove this, we are able to show that there are no limit points of
S
′′
2
of modulus less than 1.17. In so doing, we are able to produce a list of all the elements of
S
′′
2
of modulus less than 1.17, as well as a list of some of the limit points of
S
′′
2
of small modulus. All of these limit points have modulus greater than the square root of the golden ratio. In addition to this problem concerning
S
′′
2
, we will show that the totally real elements of
S
k
are limit points of
S
k
. We also give a condition under which reciprocal elements of
S
k
are limit points of
S
k
. Both of these results are generalizations of known results for the Pisot numbers.
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