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CHARACTERIZING INTEGERS AMONG RATIONAL NUMBERS WITH A UNIVERSAL-EXISTENTIAL FORMULA BJORN POONEN Abstract. We prove that Z in definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points. We also give a formula, again with universal quantifiers followed by existential quantifiers, that in any number field defines the ring of integers.
  • quaternion algebra hf
  • decision problem for exponential diophan- tine equations
  • existential formulas over z
  • intersec- tion of tf
  • ring of integers
  • rational functions
  • characteristic polynomial
  • decision problems
  • algorithm



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a dissertation
submitted to the department of physics
and the committee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
Janice Wynn Guikema
March 2004c° Copyright by Janice Wynn Guikema 2004
All Rights Reserved
Since their discovery by Bednorz and Muller˜ (1986), high-temperature cuprate
superconductors have been the subject of intense experimental research and theoret-
ical work. Despite this large-scale efiort, agreement on the mechanism of high-T hasc
not been reached. Many theories make their strongest predictions for underdoped
⁄superconductors with very low super uid density n =m . For this dissertation I im-s
newly available single crystals of very underdoped YBa Cu O (Liang et al. 1998,2 3 6+x
2002). These studies have disproved a promising theory of spin-charge separation,
measured the apparent vortex size (an upper bound on the penetration depth ‚ ),ab
and revealed an intriguing phenomenon of \split" vortices.
Scanning Hall probe microscopy is a non-invasive and direct method for magnetic
fleld imaging. It is one of the few techniques capable of submicron spatial resolution
coupled with sub-' ( ux quantum) sensitivity, and it operates over a wide tempera-0
ture range. Chapter 2 introduces the variable temperature scanning microscope and
discusses the scanning Hall probe set-up and scanner characterizations. Chapter 3
details my fabrication of submicron GaAs/AlGaAs Hall probes and discusses noise
studies for a range of probe sizes, which suggest that sub-100 nm probes could be
made without compromising ux sensitivity.
The subsequent chapters detail scanning Hall probe (and SQUID) microscopy
studies of very underdoped YBa Cu O crystals with T • 15 K. Chapter 4 de-2 3 6+x c
tion theory proposed by Senthil and Fisher (2000, 2001b). We searched for predicted
hc=e vortices (Wynn et al. 2001) and a vortex memory efiect (Bonn et al. 2001) with
vnull results, placing upper bounds on the vison energy inconsistent with the theory.
Chapter 5 discusses imaging of isolated vortices as a function of T . Vortex imagesc
size. ThedataforthelowestT ’s(5and6.5K)showsomeinhomogeneityandsuggestc
⁄that ‚ might be larger than predicted by the T /n (0)=m relation flrst suggestedab c s
by results of Uemura et al. (1989) for underdoped cuprates. Finally, Chapter 6 ex-
amines observations of apparent \partial vortices" in the crystals. My studies of
these features indicate that they are likely split pancake vortex stacks. Qualitatively,
these split stacks reveal information about pinning and anisotropy in the samples.
Collectively these magnetic imaging studies deepen our knowledge of cuprate super-
conductivity, especially in the important regime of low super uid density.
FirstandforemostIwanttothankmyadvisorKathryn(Kam)Moler. Ithasbeen
an honor to be her flrst Ph.D. student. She has taught me, both consciously and un-
consciously,howgoodexperimentalphysicsisdone. Iappreciateallhercontributions
oftime, ideas, andfundingtomakemyPh.D. experienceproductiveandstimulating.
The joy and enthusiasm she has for her research was contagious and motivational for
me,evenduringtoughtimesinthePh.D.pursuit. Iamalsothankfulfortheexcellent
example she has provided as a successful woman physicist and professor.
The members of the Moler group have contributed immensely to my personal and
professional time at Stanford. The group has been a source of friendships as well as
good advice and collaboration. I am especially grateful for the fun group of original
Moler group members who stuck it out in grad school with me: Brian Gardner, Per
Bj˜ornsson, and Eric Straver. I would like to acknowledge honorary group member
DougBonnwhowashereonsabbaticalacoupleyearsago. Weworkedtogether(along
with Brian) on the spin-charge separation experiments, and I very much appreciated
his enthusiasm, intensity, willingness to do frequent helium transfers, and amazing
ability to cleave and manipulate »50 nm crystals. Other past and present group
members that I have had the pleasure to work with or alongside of are grad students
HendrikBluhm,ClifiordHicks,Yu-JuLin,ZhifengDengandRafaelDinner; postdocs
Mark Topinka and Jenny Hofiman; and the numerous summer and rotation students
who have come through the lab.
InregardstotheHallprobes, IthankDavidKisker(formerlyatIBM),andHadas
Shtrikman at Weizmann, for growing the GaAs/AlGaAs 2DEG wafers on which the
probes were made. The Marcus group gave me advice on GaAs processes early on.
viiYu-Ju shared with me some tips she picked up during her Hall probe fab, and Clifi
rdspent a summer at Weizmann fabricating our 3 generation Hall probes. David
Goldhaber-Gordon and Mark shared some of their expert 2DEG knowledge with me.
For the noise studies, Mark wrote a spectrum analyzer program and Per, Brian, and
Rafael took some of the noise measurements. I would also like to acknowledge the
Stanford Nanofabrication Facility and the student microfabrication lab in Ginzton,
where I made the probes, and Tom Carver who did the metal evaporations.
The vortex studies discussed in this dissertation would not have been possible
withoutthehigh-puritycrystalsofunderdopedYBa Cu O fromthegroupofDoug2 3 6+x
Bonn and Walter Hardy at the University of British Columbia. I have appreciated
their collaboration and the impressive crystal growing skills of Ruixing Liang who
grew these crystals.
For the spin-charge separation tests, Doug and Brian made signiflcant contribu-
tions to the experiments, with Doug leading the way on the vortex memory experi-
ment. I also thank Matthew Fisher, Senthil Todadri, Subir Sachdev, Steve Kivelson,
Patrick Lee, Bob Laughlin and Phil Anderson for inspirational discussions with us
regarding these experiments.
In my later work of vortex fltting and studying partial vortices, I am particularly
indebted to Hendrik. He wrote the initial code to numerically generate the model
of the vortex magnetic fleld and set up the framework for fltting the model to Hall
probe images. Hendrik also performed relevant Monte Carlo simulations of thermal
motion of pancake vortices and worked out the equations describing the fleld proflles
of split pancake vortex stacks.
In my attempted measurements of the penetration depth from vortex images,
I thank the following people for helpful discussions with us: Steve Kivelson, John
Kirtley, Eli Zeldov, Aharon Kapitulnik, and Doug Bonn. For the partial vortex
work, I am especially grateful for conversations with Vladimir Kogan and also David
Santiago as we strived to determine the cause of the apparent partial vortices.
For this dissertation I would like to thank my reading committee members: Kam,
Mac Beasley, and David Goldhaber-Gordon for their time, interest, and helpful com-
ments. Iwouldalsoliketothanktheothertwomembersofmyoraldefensecommittee,
viiiShoucheng Zhang and Mark Brongersma, for their time and insightful questions.
I have appreciated the camaraderie and local expertise of the Goldhaber-Gordon
and KGB groups in the basement of McCullough, as well as the Marcus group early
on. I am grateful to our group’s administrative assistant Judy Clark who kept us
organized and was always ready to help.
IgratefullyacknowledgethefundingsourcesthatmademyPh.D.workpossible. I
andwashonoredtobeaGabilanStanfordGraduateFellowforyears4&5. Mywork
was also supported by the National Science Foundation and the U.S. Department of
groups that became a part of my life. I am grateful for time spent with roommates
for Dick and Mary Anne Bube’s hospitality as I flnished up my degree, and for many
other people and memories. My time at Stanford was also enriched by the graduate
InterVarsity group, Menlo Park Presbyterian Church, Palo Alto Christian Reformed
Church, and the Stanford Cycling Team.
Lastly, I would like to thank my family for all their love and encouragement. For
For the presence of my brother Dave here at Stanford for two of my years here. And
most of all for my loving, supportive, encouraging, and patient husband Seth whose
faithful support during the flnal stages of this Ph.D. is so appreciated. Thank you.
Janice Wynn Guikema
Stanford University
March 2004