Automorphisms of Affine Spaces: Proceedings of a Conference held in Curaçao (Netherlands Antilles), July 4–8, 1994, under auspices of the Caribbean Mathematical Foundation (CMF)

243 Pages·1995·6.06 MB·English

by  Arno van den Essen (auth.)

AUTOMORPHISMS OF AFFINE SPACES
Automorphisms of
Affine Spaces
Proceedings of a Conference held in
(Netherlands AntilIes ),
Cura~ao
July 4-8, 1994, under auspices of the
Caribbean Mathematical Foundation (CMF)
Edited by
ARNO VA N DEN ESSEN
Department ojM athematics,
Catholic University ojNijmegen,
Nijmegen, The Netherlands
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4566-9 ISBN 978-94-015-8555-2 (eBook)
DOI 10.1007/978-94-015-8555-2
Printed on acid-free paper
All Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
Softcover reprint ofthe hardcover 1st edition 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
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Table of Contents
DIRECTOR'S PREFA CE xi
FOREWORD xiii
SCIENTIFIC PROGRAM xv
Workshop .. xv
Conference . xvi
Participants . xvii
I Polynomial Maps in Dimension n 1
SEVEN LECTURES ON POLYNOMIAL
AUTOMORPHISMS 3
Arno van den Essen
1 Lecture. Invertible polynomial mapsj a first encounter 3
2 Lecture. The Jacobian Conjecture .. 8
3 Lecture. An invertibility criterion . . . . . . . . . . . . 15
4 Lecture. The tarne generators problem . . . . . . . . . 20
5 Lecture. Invertible polynomial maps and algebraic group
actions . . . . . . . . . . . . . . . . 24
5.1 The Linearization Problem 26
5.2 The Fixed Point problem . 26
5.3 The Cancelation Problem . 27
6 Lecture. The kernel of a derivation 28
7 Lecture. Why should you believe the Jacobian Conjecture? 32
8 Acknowledgments........................ 36
THE JACOBIAN CONJECTURE:
SOME STEPS TOWARDS SOLUTION 41
Ludwik Druikowski
1 Formulation of the Jacobian Conjecture 41
2 Basic facts on polynomial mappings 42
3 The Jacobian Conjecture in C2 • . • • • 44
4 Reduction of the degree . . . . . . . . . 45
5 Global asymptotic stability problem and the real Jacobian
Problem. . . . . . 46
6 Inversion formulas .. . . . . . . . . . . . . . . . . . . . .. 49
v
vi
FINITE AUTOMORPHISMS OF AFFINE N-SPACE 55
Hanspeter Kraft and Gerald Schwarz
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 56
2 The amalgamated product structure of AutA2 • • • • 56
3 Non-linearizable group actions and G-vector bundles 58
4 The equivariant SERRE-Problem for abelian groups . 61
POLYOMORPHISMS CONJUGATE TO DILATIONS 67
Gary Meisters
1 Background and terminology ...... 67
2 What's in this paper & Proposition 2.1 . 68
3 Proof of Proposition 2.1 . . . . . . . . . 69
4 Kernel matrices A for cubic-linear maps 73
5 Two cubic-linear maps showing /, /-1, h and h;l 74
s
6 Rusek's Example & Other Non-cubic-linear Examples 75
6.1 Degree 6 & 7 Schröder-terms for Rusek's 2D . . 75
6.2 Reduction of 2D-Rusek to 5D-cubic-homogeneous . 76
7 Another cubic-homogeneous but not cubic-linear 78
8 Conjectures: What seems to be true ............ 79
9 The Poincare-Siegel Theory . . . . . . . . . . . . . . . . . 80
A Nelson's example, Anick's example & Reduction of Degree. 81
A.l Nelson's Example of The Poincare-Siegel Theory .. 81
A.2 Anick's 4D cubic-homogeneous non-triangularizable 82
A.3 Reduction Algorithm . . . . . . . . . . . . . . . . .. 84
ON SEPARABLE ALGEBRAS OVER A V.F.D. AND THE
JACOBIAN CONJECTVRE IN ANY CHARACTERISTIC 89
Kossivi Adjamagbo
1 On the transfer of separability from an algebra to a sub-algebra 90
2 Isomorphism theorems far algebras over a U.F.D. 99
3 Application to the Jacobian conjecture . . . . . . . . . . .. 101
GLOBAL INJECTIVITY OF POLYNOMIAL MAPS VIA
VECTOR FIELDS 105
Anna Cima, Armengoi Gasull, Jaume LLibre and Francesc Mafiosas
1 Introduction and statement of the results 105
2 Proof of Theorem 1.2 . 108
3 Proof of Theorem 1.3. . . . . . . . . 113
4 Proof of Theorem 1.4 ........ . 118
5 On the Markus Yamabe Conjecture . 119
Scientific Program vii
11 Two-dimensional Results 125
ON THE MARKUS-YAMABE CONJECTURE 127
Robert Feßler
1 Introduction .................. . 127
2 History of the Markus - Yamabe Conjecture . 127
3 Relation to the Jacobian Conjecture. . . . . . 129
4 Solution of the two-dimensional Markus-Yamabe Conjecture 129
DERIVATIONS GENERATED BY POLYNOMIALS,
THEIR IMAGES AND COMPLEMENTS OF THE
IMAGES 137
Yosef Stein
1 Review of recent results 137
2 Some new results. . . . . 140
3 Miscellaneous questions. 144
NORMAL FORMS AND THE JACOBIAN
CONJECTURE 145
David Wright
1 Introduction . . . . . . . . . . . . . . . . . 145
2 Connection with the Jacobian Conjecture 146
3 The Intermediate Ring R . . . 146
4 Specific Coset Representatives . 147
5 Preparation Theorems . . . . . 149
6 Integrality and Exactness . . . 151
7 Reformulation of the Two-Dimensional Jacobian Conjecture 153
8 Further Observations on the Case .e = 3 . . . . . . . 154
RADIAL SIMILARITY OF NEWTON POLYGONS 157
Charles Cheng and Stuart Wang
AN ALGORITHM THAT DETERMINES WHETHER
A POLYNOMIAL MAP IS BIJECTIVE 169
Charles Cheng and Stuart Wang
1 Introduction . . . . . . . . . . 169
2 Determining KU, g] = K[t] . 169
3 Determining Automorphisms 171
4 Decomposing Automorphisms 173
viü
III Group Actions 177
ALGEBRAIC ASPECTS OF ADDITIVE GROUP ACTIONS
ON COMPLEX AFFINE SPACE 179
James Deveney and David Finston
1 Rational Triangulability . . . . . . . . . . 179
2 Rings of Ga Invariants . . . . . . . . . . . 182
3 Proper versus Locally Trivial Ga Actions . 183
4 Locally Trivial vs Equivariantly Trivial Actions 186
5 Free Ga Actions on c'3 . . . . . . . . . . . . . . 188
QUOTIENTS OF ALGEBRAIC GROUP ACTIONS 191
Harm Derksen
1 Introduction . . . . . . 191
2 Examples of quotients 191
3 Properties of actions 194
4 A counterexample .. 196
ONE-PARAMETER SUBGROUPS AND THE TRIANGULAR
SUBGROUP OF THE AFFINE CREMONA GROUP 201
Gene Freudenburg
1 One-Parameter Subgroups ..... . 202
1.1 Locally Nilpotent Derivations 202
1.2 Rank and Triangulability .. 203
1.3 Facts about Kerneis ..... 207
2 Resolving Triangular Automorphisms in Dimension 3 . 208
A NOTE ON NAGATA'S AUTOMORPHISM 215
Jacques Alev
1 Introduction . . . . . . . . . . . . . . . . . . . . 215
2 The tameness problem and Nagata's automorphism. 215
3 The main observation ................ . 217
IV Reactions on the conference 223
ON A QUESTION OF YOSEF STEIN 225
H anspeter K rojt
A COUNTEREXAMPLE TO A CONJECTURE OF
MEISTERS 231
Amo van den Essen
1 The counterexample 231
Scientific Program ix
2 Meisters Conjecture holds if n ~ 3 232
3 Acknowledgment . . . . . . . . . . 233
Appendix 235
OPEN PROBLEMS 237
SOME CONFERENCE IMPRESSIONS 241
DIRECTOR'S PREFACE
The reader is presented with this re cord of the workshop/conference on
Invertible Polynomial Maps, which was held in Curac;ao, Netherlands An
tilles, from the 1st through the 9th of July, 1994, under the auspices of the
Caribbean Mathematics Foundation (abbr. CMF). It is a pleasure to see
this project brought to completion with such a splendid document.
The aforementioned conference has served to showcase the raison d'etre
of CMF. For this event, a group of some 20 mathematicians of the high
est caliber gathered in Curac;ao, to discuss one of the most classieal open
problems in mathematics: the Jacobian Conjecture and its peripheries. The
group included the expected proportion of participation by mathematicians
from North America and Europe, but also mathematicians from Jamaica,
Surinam, Venezuela and Curac;ao itself. On the 1st & 2nd of July, the or
ganizer, Professor Arno van den Essen (Nijmegen) presented aseries of
expository lectures on the Jacobian Conjecture. These lectures were well
motivated, from a historieal consideration, and it was amply demonstrated
to the mathematician in the audience, who was not expert in the area, that
the problem has many faces, and multiple applications, as weIl as modes
of approach. It is truly a research discipline which has something to offer
to mathematicians of vastly different persuasions, and Professor van den
Essen's expositions brought this out with clarity.
CMF was fortunate to have Professor van den Essen organize this con
ference. He understood, from the start of this project, and instinctively,
wh at was required for the event, in terms of balance of exposition and
"state-of-the-art" presentations. Professor van den Essen has the kind of
"missionary" spirit which practically guarantees success in ventures such
as this one. His organization made my job as broker for these events the
pleasure I always hope for.
This event was supported by the CMF, which in turn owes its financial back
ing to the generous contributions by the European Economie Community,
the University of the Netherlands Antilles, the University of Florida, and
members of the private sector of the island, to wit: the Maduro & Curiel's
Bank, AHOLD Insurance & AHOLD Finance Corp., CITCO Trust, KMPG
Associates, Mees Pierson, Refinerfa ISLA and Refinerfa di Korsou.
The contributions just mentioned here did not fiow to the participants
by magie. I am fortunate to have had the friendship and cooperation of
several individuals who made it aIl possible. In making the financial sup
port of the European Community a reality, I was blessed by having the
unitiring support of Ms. Lygia SteIla of the Department of Development
xi
xü
& Cooperation in Cura~ao; without her diligence I (and perhaps others)
would have lost their sanity.
In my department we have Ms. JoAnne McLeary, whose task (among
others) is to make sure that travel funds are paid on time. With regard to
these CMF gatherings, which do not take place on the University of Florida
campus, her efforts are all the more heroic. How wonderful to be able to
count Ms. McLeary's among my (obviously lucky) stars.
In Cura~ao there is a distinguished gentleman named Huub Bongers,
who taught mathematics to several generations of high school students;
a man of boundless optimism and (happily) bountiful resources in the en
trepreneurial community. He has acted as the happy catalyst, which brought
about the contributions by that community of the island.
As always, I have been fortunate to have the support of many individ
uals associated with the University of the Netherlands Antilles, and in this
regard, I should like to recognize two people in particular, this year. First,
the man who always pays the bills, and from whom I have come to expect
such efficiency that I almost fail to notice hirn. Mr. R. Ravenstein and I
rarely meet, as he has such a busy schedule, and this past summer I did not
see hirn at ail. We have cordial exchanges by fax, and somehow everything
happens when it is supposed to, without fail.
Second, I am enriched indeed by the friendship of Dr. EImer Joubert,
Rector Magnificus of the University of the Netherlands Antilles. The office
is graced by his tenure, and I am grateful for his conversation and his sound
advice.
Finaily, I am grateful to Kluwer Academic Publishers, to Dr. David
Larner, Ms. Margaret Deignan and the entire staff of the Science & Tech
nology Division, for their many kindnesses and their support, and to the
publisher, as weil, for its generosity in making 20 copies of this volume
available, free of charge, for distribution in the Caribbean Region and in
Latin America.
No smail surprise then, that I welcome the completion of these proceedings,
which round out the conference event rat her smartly, and do ample justice
to the mathematicians who participated in it and to their craft.
Jorge Martinez, Director
CARIBBEAN MATHEMATICS FOUNDATION
Gainesville, Florida
October, 1994