Table Of ContentCMS/CAIMS Books in Mathematics
Canadian
Mathematical Society
Société mathématique
Pinaki Mondal du Canada
How Many
Zeroes ?
Counting Solutions of Systems
of Polynomials via Toric Geometry
at Infinity
CMS/CAIMS Books in Mathematics
Volume 2
Series Editors
Karl Dilcher
Dalhousie University, Halifax, Canada
Frithjof Lutscher
University of Ottawa, Ottawa, Canada
Nilima Nigam
Simon Fraser University, Burnaby, Canada
Keith Taylor
Dalhousie University, Halifax, Canada
Associate Editors
Ben Adcock
Simon Fraser University, Burnaby, Canada
Martin Barlow
University of British Columbia, Vancouver, Canada
Heinz H. Bauschke
University of British Columbia, Kelowna, Canada
Matt Davison
Western University, London, Canada
Leah Keshet
University of British Columbia, Vancouver, Canada
Niky Kamran
McGill University, Montreal, Canada
Mikhail Kotchetov
Memorial University of Newfoundland, St. John’s, Canada
Raymond J. Spiteri
University of Saskatchewan, Saskatoon, Canada
CMS/CAIMS Books in Mathematics is a collection of monographs and graduate-
level textbooks published in cooperation jointly with the Canadian Mathematical
Society-SocietémathématiqueduCanadaandtheCanadianAppliedandIndustrial
Mathematics Society-Societé Canadienne de Mathématiques Appliquées et Indus-
trielles.Thisseries offers authorsthejoint advantageofpublishingwithtwomajor
mathematical societies and with a leading academic publishing company. The se-
ries is edited by Karl Dilcher, Frithjof Lutscher, Nilima Nigam, and Keith Taylor.
The series publishes high-impact works across the breadth of mathematics and its
applications. Books in this series will appeal to all mathematicians, students and
established researchers. The series replaces the CMS Books in Mathematics series
that successfully published over 45volumes in20 years.
More information about this series at http://www.springer.com/series/16627
Pinaki Mondal
How Many Zeroes?
Counting Solutions of Systems
fi
of Polynomials via Toric Geometry at In nity
123
Pinaki Mondal
Scarborough,ON, Canada
ISSN 2730-650X ISSN 2730-6518 (electronic)
CMS/CAIMS Booksin Mathematics
ISBN978-3-030-75173-9 ISBN978-3-030-75174-6 (eBook)
https://doi.org/10.1007/978-3-030-75174-6
MathematicsSubjectClassification: 14C17,14M25,52B20,14N10
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The publisher, the authors and the editors are safe to assume that the advice and information in this
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To my parents Purnima Mondal and Monojit Mondal
Preface
Inthisbookwedescribeanapproachthrough toricgeometrytothefollowing
problem: “estimate the number (counted with appropriate multiplicity) of isolated
solutionsofnpolynomialequationsinnvariablesoveranalgebraicallyclosedfield
k.”Theoutcomeofthisapproachisthenumberofsolutionsfor“generic”systems
intermsoftheir Newtonpolytopes,andanexplicitcharacterizationofwhatmakes
a system “generic.” The pioneering work in this field was done in the 1970s by
Kushnirenko, Bernstein and Khovanskii, who completely solved the problem of
countingsolutionsofgenericsystemsonthe“torus”ðknf0gÞn.Inthecontextofour
problem, however, the natural domain of solutions is not the torus, but the affine
space kn. There were a number of works on extending Bernstein’s theorem to the
case of affine space, and recently it has been completely resolved, the final steps
having been carried out by the author.
Theaimofthisbookistopresenttheseresultsinacoherentway.Westartfrom
the beginning, namely, Bernstein’s beautiful theorem which expresses the number
of solutions of generic systems on the torus in terms of the mixed volume of their
Newton polytopes. We give complete proofs, over arbitrary algebraically closed
fields, of Bernstein’s theorem, its recent extension to the affine space, and some
other related applications including generalizations of Kushnienko’s results on
Milnornumbersofhypersurfacesingularitieswhichin1970sservedasaprecursor
to the development of toric geometry. Our proofs of all these results share several
key ideas and are accessible to someone equipped with the knowledge of basic
algebraicgeometry.Thisbookcanserveasacompaniontointroductorycourseson
algebraic geometry or toric varieties. While it does not provide a comprehensive
introductiontoalgebraicgeometry,itdoesdeveloptherelevantpartsofthesubject
from the beginning (modulo some explicitly stated basic results) with lots of
examples and exercises and can be used as a quick introduction to basic algebraic
geometry. We hope the readers who take that undertaking will be rewarded by a
deep understanding of the affine Bézout problem.
vii
viii Preface
Acknowledgements. ItwasPierreMilman whowanted metowriteabook; it
would not have been possible without his constant encouragement and support—
with mathematics, and all sorts of things beyond mathematics—throughout these
years.Eventhoughthescopeofthefinalversionisconsiderablylimitedcompared
to his vision, I offer it as a first step. The encouragement from Eriko Hironaka
workedasacatalystduringacriticalperiodwhentheprojectwasstuck.Isincerely
thank Jan Stevens who sent numerous corrections after reading one of the earlier
drafts. Najma Ahmad, Kinjal Dasbiswas, Naren Hoovinakatte, and especially,
Jonathan Korman read parts of earlier drafts and gave important suggestions.
Thanks are also due to the referees and editors, especially Keith Taylor, whose
suggestionssignificantlyimprovedtheexposition.Overthelastfewyearsthework
on this book took a great portion of my time owed to my friends and family,
especiallymymotherPurnimaMondalandbrotherProtimMondol.Theapplication
ofpointsatinfinitytoChickens’RoadCrossingproblemisduetoShatabdiSarker;
its presentation given in this book is due to Tanzil Rashid.
Scarborough, Canada Pinaki Mondal
Contents
Chapter I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. The problem and the results. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3. Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Chapter II. A brief history of points at infinity in geometry . . . . . . . . . 15
1. Points at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2. Homogeneous coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3. Projective space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Part 1. Preliminaries
Chapter III. Quasiprojective varieties over algebraically
closed fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1. Affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2. (Ir)reducibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3. Regular functions, coordinate rings and morphisms of affine
varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4. Quasiprojective varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5. Regular functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6. Morphisms of quasiprojective varieties; affine varieties as
quasiprojective varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7. Rational functions and rational maps on irreducible varieties . . . . 48
7.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
ix
x Contents
8. Product spaces, Segre map, Veronese embedding . . . . . . . . . . . . 51
8.1. Product spaces, Segre map . . . . . . . . . . . . . . . . . . . . . . . . 51
8.2. Veronese embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.3. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
9. Completeness and compactification. . . . . . . . . . . . . . . . . . . . . . . 55
9.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
10. Image of a morphism: Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
10.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
11. Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
11.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
12. Image of a morphism: Part II - Constructible sets . . . . . . . . . . . . 69
12.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
13. Tangent space, singularities, local ring at a point. . . . . . . . . . . . . 72
13.1. The case of affine varieties . . . . . . . . . . . . . . . . . . . . . . . . 72
13.2. Intrinsicness of the tangent space; tangent spaces and
singularities on arbitrary varieties . . . . . . . . . . . . . . . . . . . 73
13.3. Equations near a nonsingular point . . . . . . . . . . . . . . . . . . 76
13.4. Parametrizations of a curve at a nonsingular point . . . . . . . 77
13.5. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
14. Completion of the local ring at a point . . . . . . . . . . . . . . . . . . . . 81
14.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
15. Degree of a dominant morphism. . . . . . . . . . . . . . . . . . . . . . . . . 84
Chapter IV. (cid:2)Intersection multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . 89
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2. Closed subschemes of a variety . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.1. Closed subschemes of an affine variety . . . . . . . . . . . . . . . 90
2.2. Closed subschemes of a quasiprojective variety . . . . . . . . . 91
2.3. Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2.4. Completeness and compactification of schemes . . . . . . . . . 94
2.5. Irreducible components, local rings . . . . . . . . . . . . . . . . . . 96
2.6. Cartier divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3. Possibly non-reduced curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.1. (Reduced) Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.2. Order at a point on a possibly non-reduced curve . . . . . . . 99
4. Intersection multiplicity at a nonsingular point of a variety . . . . . 102
4.1. Intersection multiplicity of power series . . . . . . . . . . . . . . 102
4.2. Intersection multiplicity of regular functions . . . . . . . . . . . 103
4.3. Intersection multiplicity in a family. . . . . . . . . . . . . . . . . . 104
5. Intersection multiplicity of complete intersections . . . . . . . . . . . . 108
Chapter V. Convex polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
1. Basic notions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
1.1. Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
