Real solutions to equations from geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2011
|
| Series: | University lecture series
57 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | IX, 200 S. graph. Darst. |
| ISBN: | 9780821853313 |
Staff View
MARC
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| 245 | 1 | 0 | |a Real solutions to equations from geometry |c Frank Sottile |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2011 | |
| 300 | |a IX, 200 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804148538973618176 |
|---|---|
| adam_text | Contents
Preface
Chapter
1.
Overview
1
1.1.
Introduction
2
1.2.
Polyhedral bounds
3
1.3.
Upper bounds
4
1.4.
The
Wroński
map and the Shapiro Conjecture
5
1.5.
Lower bounds
8
Chapter
2.
Real Solutions to
Uni variate
Polynomials
13
2.1.
Descartes s rule of signs
13
2.2.
Sturm s Theorem
16
2.3.
A topological proof of Sturm s Theorem
19
Chapter
3.
Sparse Polynomial Systems
25
3.1.
Polyhedral bounds
26
3.2.
Geometric interpretation of sparse polynomial systems
27
3.3.
Proof of Kushnirenko s Theorem
29
3.4.
Facial systems and degeneracies
33
Chapter
4.
Toric Degenerations and Kushnirenko s Theorem
37
4.1.
Kushnirenko s Theorem for a simplex
37
4.2.
Regular subdivisions and toric degenerations
39
4.3.
Kushnirenko s Theorem via toric degenerations
44
4.4.
Polynomial systems with only real solutions
47
Chapter
5.
Fewnomial Upper Bounds
49
5.1.
Khovanskii s fewnomial bound
49
5.2.
Kushnirenko s Conjecture
54
5.3.
Systems supported on a circuit
56
Chapter
6.
Fewnomial Upper Bounds from Gale Dual Polynomial Systems
61
6.1.
Gale duality for polynomial systems
62
6.2.
New fewnomial bounds
66
6.3.
Dense fewnomials
74
Chapter
7.
Lower Bounds for Sparse Polynomial Systems
77
7.1.
Polynomial systems as fibers of maps
78
7.2.
Orientability of real toric varieties
80
7.3.
Degree from foldable
triangulations
84
7.4.
Open problems
89
viii CONTENTS
Chapter
8.
Some Lower Bounds for Systems of Polynomials
91
8.1.
Polynomial systems from posets
91
8.2.
Sagbi degenerations
96
8.3.
Incomparable chains, factoring polynomials, and gaps
100
Chapter
9.
Enumerative Real Algebraic Geometry
105
9.1. 3264
real conies
105
9.2.
Some geometric problems
109
9.3.
Schubert Calculus
116
Chapter
10.
The Shapiro Conjecture for Grassmannians
121
10.1.
The
Wroński
map and Schubert Calculus
122
10.2.
Asymptotic form of the Shapiro Conjecture
124
10.3. Grassmann
duality
130
Chapter
11.
The Shapiro Conjecture for Rational Functions
133
11.1.
Nets of rational functions
133
11.2.
Schubert induction for rational functions and nets
137
11.3.
Rational functions with prescribed coincidences
141
Chapter
12.
Proof of the Shapiro Conjecture for Grassmannians
147
12.1.
Spaces of polynomials with given Wronskian
148
12.2.
The Gaudin model
152
12.3.
The Bethe
Ansatz
for the Gaudin model
154
12.4.
Shapovalov form and the proof of the Shapiro Conjecture
157
Chapter
13.
Beyond the Shapiro Conjecture for the Grassmannian
161
13.1.
Transversality and the Discriminant Conjecture
161
13.2.
Maximally inflected curves
164
13.3.
Degree of
Wroński
maps and beyond
167
13.4.
The Secant Conjecture
170
Chapter
14.
The Shapiro Conjecture Beyond the Grassmannian
173
14.1.
The Shapiro Conjecture for the orthogonal Grassmannian
173
14.2.
The Shapiro Conjecture for the Lagrangian Grassmannian
175
14.3.
The Shapiro Conjecture for flag manifolds
179
14.4.
The Monotone Conjecture
180
14.5.
The Monotone Secant Conjecture
186
Bibliography
189
Index of Notation
195
Index
197
Understanding, finding, or even deciding on the existence of real
solutions to a system of equations is a difficult problem with many
applications outside of mathematics. While it is hopeless to expect
much in general, we know a surprising amount about these questions
for systems which possess additional structure often coming from
geometry.
This book focuses on equations from tone varieties and Grassmannians.
Not only is much known about these, but such equations are common
in applications. There are three main themes: upper bounds on the
number of real solutions, lower bounds on the number of real solutions, and geometric
problems that can have all solutions be real. The book begins with an overview, giving
background on real solutions to univariate polynomials and the geometry of sparse poly¬
nomial systems. The first half of the book concludes with fewnomial upper bounds and
with lower bounds to sparse polynomial systems. The second half of the book begins by
sampling some geometric problems for which all solutions can be real, before devoting the
last five chapters to the Shapiro Conjecture, in which the relevant polynomial systems have
only real solutions.
For additional information
and updates on this book, visit
www.ams.org/bookpages/ulect-S?
ISBN
978-0-8218-5331-3
9ІІ780821 853313І
ULECT/57
|
| any_adam_object | 1 |
| author | Sottile, Frank 1963- |
| author_GND | (DE-588)1017432406 |
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| dewey-full | 516.3/53 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/53 |
| dewey-search | 516.3/53 |
| dewey-sort | 3516.3 253 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-10T00:08:44Z |
| institution | BVB |
| isbn | 9780821853313 |
| language | English |
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| physical | IX, 200 S. graph. Darst. |
| publishDate | 2011 |
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| publisher | American Math. Soc. |
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| series | University lecture series |
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| spelling | Sottile, Frank 1963- (DE-588)1017432406 aut Real solutions to equations from geometry Frank Sottile Providence, RI American Math. Soc. 2011 IX, 200 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier University lecture series 57 Algebraische Varietät (DE-588)4581715-7 gnd rswk-swf Algebraische Varietät (DE-588)4581715-7 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1652-2 University lecture series 57 (DE-604)BV004153846 57 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024520752&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024520752&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Sottile, Frank 1963- Real solutions to equations from geometry University lecture series Algebraische Varietät (DE-588)4581715-7 gnd |
| subject_GND | (DE-588)4581715-7 |
| title | Real solutions to equations from geometry |
| title_auth | Real solutions to equations from geometry |
| title_exact_search | Real solutions to equations from geometry |
| title_full | Real solutions to equations from geometry Frank Sottile |
| title_fullStr | Real solutions to equations from geometry Frank Sottile |
| title_full_unstemmed | Real solutions to equations from geometry Frank Sottile |
| title_short | Real solutions to equations from geometry |
| title_sort | real solutions to equations from geometry |
| topic | Algebraische Varietät (DE-588)4581715-7 gnd |
| topic_facet | Algebraische Varietät |
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