Algebraic and combinatorial properties of certain toric ideals in theory and applications

75 Pages·2016·0.518 MB·English

by  Sonja Petrovic

University of Kentucky
UKnowledge
University of Kentucky Doctoral Dissertations Graduate School
2008
ALGEBRAIC AND COMBINATORIAL
PROPERTIES OF CERTAIN TORIC IDEALS
IN THEORY AND APPLICATIONS
Sonja Petrovic
University of Kentucky, [email protected]
Recommended Citation
Petrovic, Sonja, "ALGEBRAIC AND COMBINATORIAL PROPERTIES OF CERTAIN TORIC IDEALS IN THEORY AND
APPLICATIONS" (2008).University of Kentucky Doctoral Dissertations.Paper 606.
http://uknowledge.uky.edu/gradschool_diss/606
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ABSTRACT OF DISSERTATION
Sonja Petrovi´c
The Graduate School
University of Kentucky
2008
ALGEBRAIC AND COMBINATORIAL PROPERTIES OF CERTAIN TORIC
IDEALS IN THEORY AND APPLICATIONS
ABSTRACT OF DISSERTATION
A dissertation submitted in partial fulfillment of the
requirements of the degree of Doctor of Philosophy in the
College of Arts and Sciences at the University of Kentucky
By
Sonja Petrovi´c
Lexington, Kentucky
Director: Dr. Uwe Nagel, Department of Mathematics
Lexington, Kentucky
2008
ABSTRACT OF DISSERTATION
ALGEBRAIC AND COMBINATORIAL PROPERTIES OF CERTAIN TORIC
IDEALS IN THEORY AND APPLICATIONS
This work focuses on commutative algebra, its combinatorial and computational
aspects, and its interactions with statistics. The main objects of interest are pro-
jective varieties in Pn, algebraic properties of their coordinate rings, and the combi-
natorial invariants, such as Hilbert series and Gr¨obner fans, of their defining ideals.
Specifically, the ideals in this work are all toric ideals, and they come in three flavors:
they are defining ideals of a family of classical varieties called rational normal scrolls,
cut ideals that can be associated to a graph, and phylogenetic ideals arising in a new
and increasingly popular area of algebraic statistics.
Sonja Petrovi´c
May 2008
ALGEBRAIC AND COMBINATORIAL PROPERTIES OF CERTAIN TORIC
IDEALS IN THEORY AND APPLICATIONS
By
Sonja Petrovi´c
Uwe Nagel
(Director of Dissertation)
Qiang Ye
(Director of Graduate Studies)
May 2008
(Date)
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Name and Address Date
DISSERTATION
Sonja Petrovi´c
The Graduate School
University of Kentucky
2008
ALGEBRAIC AND COMBINATORIAL PROPERTIES OF CERTAIN TORIC
IDEALS IN THEORY AND APPLICATIONS
DISSERTATION
A dissertation submitted in partial fulfillment of the
requirements of the degree of Ph.D.
at the University of Kentucky
By
Sonja Petrovi´c
Lexington, Kentucky
Director: Dr. Uwe Nagel, Department of Mathematics
Lexington, Kentucky
2008
ACKNOWLEDGMENT
”The two operations of our understanding, intuition and deduction, on which
alone we have said we must rely in the acquisition of knowledge.”
–Rene Descartes
This work has benefited greatly from the teachings and insights of my adviser,
Uwe Nagel. I am deeply grateful for his endless support, dedication, and motivation
to pursue the research program described in this dissertation. He has instilled in
me a curiosity which does not end with this work. In addition, I am indebted to
Alberto Corso for his continuous guidance and support during the past few years. I
would also like to thank the rest of the Dissertation Committee, Arne Bathke, David
Leep, and the outside examiner, Kert Viele, for the time they have spent on this
dissertation and its defense.
The five years I spent at the University of Kentucky would not have been so
delightfulwithoutmydearfriendandcollaborator, JuliaChifman, withwhomIhave
shared research projects, teaching problems, office space, and many mathematical
ideas. TherearemanymorepeopletowhomIamgratefulformotivatingdiscussions,
of which I must single out Bernd Sturmfels, Seth Sullivant, and Ruriko Yoshida.
I would not be where I am today without the love and support of my family,
relatives, and friends. I owe everything to Mama, Tata and Bojan, for it is their
sacrificeandpositiveenergyduringthehardestoftimesthathavemademyeducation
possible. They have inspired me to take every opportunity that knocks on my door.
And last, but not least, I am most grateful to Saˇsa for always being there to pick me
up when I fall, for helping me believe in myself, and for teaching me that no matter
what happens today, we should never loose hope for a better tomorrow.
As a wise man once said, the world is round and the place which may seem like
the end may also be only the beginning.
iii
Contents
List of Files v
1 Introduction 1
1.1 Universal Gr¨obner bases of rational normal scrolls . . . . . . . . . . . 1
1.2 Toric ideals of phylogenetic invariants . . . . . . . . . . . . . . . . . . 2
1.3 Cut ideals of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Background 5
2.1 Ideals and Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Gr¨obner bases of toric ideals . . . . . . . . . . . . . . . . . . . . . . . 9
3 Rational normal scrolls 14
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Parametrization of Scrolls . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Colored partition identities and Graver bases . . . . . . . . . . . . . . 18
3.4 Degree bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Universal Gr¨obner bases . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Phylogenetic ideals 27
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Matrix representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Number of lattice basis elements . . . . . . . . . . . . . . . . . . . . . 31
4.4 Lattice basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5 Ideal of invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.5.1 The tree on 3 leaves . . . . . . . . . . . . . . . . . . . . . . . 35
4.5.2 The tree on an arbitrary number of leaves . . . . . . . . . . . 36
5 Cut ideals 42
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2 Clique sums, Segre products, Gr¨obner bases . . . . . . . . . . . . . . 43
5.3 Cut ideals of cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4 Cut ideals of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.5 Disjoint unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.6 Cut ideals of series-parallel graphs . . . . . . . . . . . . . . . . . . . . 56
Bibliography 61
Vita 64
iv