The most comprehensive treatment to date has been Dieudonne [11], a seventy-
one page monograph devoted exclusively to our subject.
Though very excellent, these surveys have been handicapped by a lack of
the space required for an adequate treatment of the subject. There still remains
the need for a detailed exposition which would bring together results at present
scattered throughout the mathematical journals and which would endeavor to
unify and to simplify both the results and the methods of treatment.
The present book is an attempt to fill this need. In it an effort will be made
to present the subject as completely as possible within the allotted space. Some
of the results which could not be included in the main text have been listed as
exercises, with occasional hints as to how they may be derived by use of the
material in the main text. In addition, our bibliography refers each listed paper
to the section of our text containing the material most closely allied to that in
the paper, whether or not an actual reference to that paper is made in our text.
It is hoped that this book will serve the present and prospective specialist
in the field by acquainting him with the current state of knowledge in the various
phases of the subject and thus by helping him to avoid in the future the duplica-
tion of results which has occurred all too frequently in the past. It is hoped
also that this book will serve the applied mathematician and engineer who need
to know about the distribution of the zeros of polynomials when dealing with
such matters as the formulation of stability criteria. Finally, it is hoped that
this book will serve the general mathematical reader by introducing him to some
relatively new, interesting and significant material of geometric nature—material
which, though derived by essentially elementary methods, is not readily available
In closing, the author wishes to express his deep gratitude to Professor Joseph
L. Walsh of Harvard University for having initiated the author into this field
and for having encouraged his further development in it; also, for having made
many helpful criticisms and suggestions concerning the present manuscript.
The author wishes to acknowledge his indebtedness to The University of Wisconsin
in Milwaukee for providing the assistance of Francis J. Stern in typing the
manuscript and of Richard E. Barr, Jr. in drawing most of the accompanying
figures; also his indebtedness to his colleagues at Madison for the opportunity
of giving there, from February to June 1948, a course of lectures based upon the
material in this book. Last but not least, the author wishes to thank the
American Mathematical Society for granting him the privilege of publishing this
manuscript in the Mathematical Surveys Series.
Milwaukee, Wisconsin
November 1, 1947
and October 1, 1948. MORRIS MARDEN
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