Weighted Expansions for Canonical Desingularization
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910
Shreeram S. Abhyankar
Weighted Expansions for Canonical Desingularization With Foreword by U. Orbanz
Springer-Verlag Berlin Heidelberg New York 1982
Author
Shreeram S. Abhyankar Purdue University, Div. Math. Sci. West Lafayette, IN 47907, USA
AMS Subject Classifications (1980): 14E15
ISBN 3-540-11195-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11195-6 Springer-Verlag New York Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Table of contents Section
Page
Foreword
v
Preface .
1
Notation.
3
§ 2.
Semigroups
4
§ 3.
Strings .
5
§
1.
§ 4.
Semigroup strings with restrictions
§ 5.
Ordered semigroup strings with restrictions
10
§ 6.
Strings on rings
11
7
§ 7.
Indeterminate strings
14
§ 8.
Indeterminate strings with restrictions
21
§ 9.
Restricted degree and order for indeterminate strings
26
§10.
Indexing strings.
29
§11.
Nets
31
§12.
Semigroup nets with restrictions
33
§13.
Ordered semigroup nets with restrictions
36
§14.
Nets on rings
37
§15.
Indeterminate nets
39
§16.
Indeterminate nets with restrictions
46
§17.
Restricted degree and order for indeterminate nets.
53
§18.
Prechips.
••••.
57
§19.
Isobars for prechips and Premonic polynomials
59
§20.
Substitutions • • • • • .
• •
67
§21.
Substitutions with restrictions
73
§22.
Coordinate nets and Monic polynomials
82
§23.
Graded ring of a ring at an ideal
85
§24.
Graded ring of a ring
88
§25.
Graded rings at strings and nets and the notions of separatedness and regularity for strings and nets.
90
§26.
Inner products and further notions of separatedness and regularity for strings . . . . . • . • • • • .
104
§27.
Inner products and further notions of senaratedness and regularity for nets • • • . • . •
109
§28.
Weighted isobars and weighted initial forms • • • •
113
•
. .
IV
Initial forms for regular strings . • • • •
126
§30.
Initial forms for regular strings and nets
150
§31.
Protochips and parachips • • • • • • • • .
161
§32.
N-support of an indexing string for ?
162
§33.
Prescales • •
§34.
Derived pres cales
165
§35.
Supports of prescales
167
§36.
Protoscales . .
168
§37.
Inner products for protoscales
170
§38.
Scales and isobars .
171
§39.
Properties of derived prescales
176
§40.
Isobars for derived scales .
203
§41.
Isobars and initial forms for scales
205
§42.
Initial forms for scales and regular nets
214
§43.
Isobars for protochips .
220
§44.
Initial forms for protochips a
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