University of Kentucky UKnowledge Theses and Dissertations--Mathematics Mathematics 2021 The Smallest Solution of an Isotropic Quadratic Form Deborah H. Blevins University of Kentucky, [email protected] Author ORCID Identifier: https://orcid.org/0000-0002-6815-5290 Digital Object Identifier: https://doi.org/10.13023/etd.2021.173 Right click to open a feedback form in a new tab to let us know how this document benefits ou.y Recommended Citation Blevins, Deborah H., "The Smallest Solution of an Isotropic Quadratic Form" (2021). Theses and Dissertations--Mathematics. 80. https://uknowledge.uky.edu/math_etds/80 This Doctoral Dissertation is brought to you for free and open access by the Mathematics at UKnowledge. It has been accepted for inclusion in Theses and Dissertations--Mathematics by an authorized administrator of UKnowledge. For more information, please contact [email protected]. STUDENT AGREEMENT: I represent that my thesis or dissertation and abstract are my original work. Proper attribution has been given to all outside sources. I understand that I am solely responsible for obtaining any needed copyright permissions. I have obtained needed written permission statement(s) from the owner(s) of each third-party copyrighted matter to be included in my work, allowing electronic distribution (if such use is not permitted by the fair use doctrine) which will be submitted to UKnowledge as Additional File. I hereby grant to The University of Kentucky and its agents the irrevocable, non-exclusive, and royalty-free license to archive and make accessible my work in whole or in part in all forms of media, now or hereafter known. I agree that the document mentioned above may be made available immediately for worldwide access unless an embargo applies. I retain all other ownership rights to the copyright of my work. I also retain the right to use in future works (such as articles or books) all or part of my work. I understand that I am free to register the copyright to my work. REVIEW, APPROVAL AND ACCEPTANCE The document mentioned above has been reviewed and accepted by the student’s advisor, on behalf of the advisory committee, and by the Director of Graduate Studies (DGS), on behalf of the program; we verify that this is the final, approved version of the student’s thesis including all changes required by the advisory committee. The undersigned agree to abide by the statements above. Deborah H. Blevins, Student Dr. David Leep, Major Professor Dr. Benjamin Braun, Director of Graduate Studies The Smallest Solution of an Isotropic Quadratic Form DISSERTATION A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the College of Arts and Sciences at the University of Kentucky By Deborah H. Blevins Lexington, Kentucky Director: Dr. David Leep, Professor of Mathematics Lexington, Kentucky 2021 Copyright© Deborah H. Blevins 2021 https://orcid.org/0000-0002-6815-5290 ABSTRACT OF DISSERTATION The Smallest Solution of an Isotropic Quadratic Form Pn Pn An isotropic quadratic form f(x1; : : : ; xn) = i=1 j=1 fijxixj defined on a Z- lattice has a smallest solution, where the size of the solution is measured using the infinity norm (k k1), the `1 norm (k k1), or the Euclidean norm (k k2). Much work has been done to find the least upper bound and greatest lower bound on the smallest solution, beginning with Cassels in the mid-1950's. Defining F := (f11; : : : ; f1n; f21; : : : ; f2n; : : : ; fn1; : : : ; fnn), upper bound results have the form kxki ≤ θ CkFki , with i 2 f1; 2; 1g and C a constant depending only on n. Aside from Cassels and Davenport, authors have concentrated more on finding the smallest exponent θ and less on C. Since Cassels' publication, others have generalized his result and an- swered related questions. In particular, Schlickewei and Schmidt considered cases in which a quadratic form vanishes on a space of dimension greater than 1 and whether or not there is a bound on the product of the norms of multiple linearly independent nontrivial solutions. Schulze-Pillot explored similar questions, in addition to consider- ing bounds on the determinant of the matrix whose columns are linearly independent nontrivial solutions of a quadratic form. Another goal has been to determine the best possible bound in each of these cases. In particular, Kneser demonstrated Cassels' bound is best possible and Schlickewei and Schmidt have shown their results, along with several of Schulze-Pillot's, are best possible. This dissertation gives a detailed, comprehensive exposition of these results. In particular, many of the results of Schulze-Pillot and Schlickewei and Schmidt are re- examined, leading to greater insight into the details of the proofs. The main methods used to solve this problem come from quadratic form theory, lattice theory, and the geometry of numbers. KEYWORDS: isotropic quadratic forms, lattice, geometry of numbers, successive minima Deborah H. Blevins May 12, 2021 The Smallest Solution of an Isotropic Quadratic Form By Deborah H. Blevins Dr. David Leep Director of Dissertation Dr. Benjamin Braun Director of Graduate Studies May 12, 2021 Date \Therefore I run thus: not with uncertainty. Thus I fight: not as one who beats the air." 1 Corinthians 9:26 To Logan, Daddy, Mama, and Daniel { Thank you for running the race with me and teaching me what matters most. ACKNOWLEDGMENTS First and foremost I would like to thank Dr. David Leep, whose guidance, direction, and support made this possible. After our first conversation, my impression was that you were kind and thoughtful. The years since have proven this to be true over and over. Thank you for the many hours you have spent helping me, telling me stories, making me laugh, and for your endless advice { both about mathematics and about life. I am a better mathematician, and a better human, for it. I am deeply grateful to the remaining members of my committee { Drs. Corso, Shao, Srinivasan, and Jaromczyk { for your support and commitment. Thank you to Dr. Ben Braun, our Director of Graduate Studies, for helping me decide which prelims to take when I was in despair after so many failed attempts and for always responding kindly to panicked emails. Thank you to Dr. erica Whitaker, our Director of Service Courses, for advocating for graduate students, helping me grow as a teacher, always having your door open, and having candy in your office. Thank you to Christine Levitt, Rejeana Cassady, and Sheri Rhine for holding our department together, and each of us along with it. Dr. Manning, Dr. Newquist, and Dr. Ramey { thank you for believing I could do this before it was a thought in my mind. You all showed me what it looks like to invest in your students both inside and outside the classroom and pushed me to be a leader when I didn't want to be. I hope to be like you. I would also like to thank Dr. Robert Bridges for giving me the opportunity to participate in the NSF Mathematical Sciences Graduate Internship at Oak Ridge National Laboratory. What I thought would be a ten-week internship turned into a much longer collaboration and richer experience than I could have anticipated. You have gone above and beyond in mentorship and I am so grateful. Thank you also iii to Miki Verma { for making a space in your office for me, for answering a thousand questions, and for understanding my need for a regular schedule. To my dear husband, Logan, thank you for being brave enough to marry a grad- uate student, for believing in me more than I believe in myself, for keeping me moti- vated when my tank is empty, and for being excited about the next adventure. Your steadiness has been my life raft the past three years. This is all better with you by my side. To my parents and brother, David, Donna, and Daniel Wilkerson, there are not words adequate enough to express my gratitude nor enough space to list all the ways you've encouraged and supported me. You are my original support system, my first phone call, and the reason I am in graduate school. Stephen and Kathy Blevins { thank you for caring for me as your own, for asking all the questions in order to understand what I do in graduate school, and for indulging my affinity for infinity. You all are such a precious addition to my life and to this support team. My time as a graduate student would have come to a quick end if it were not for the most spectacular collection of friends I made in the department. Kalila, Julie, Darleen, Kevin, and Rachel P. { thank you for cooking me so many meals, going for walks, laughing with me, crying with me, and cheering me on. Thank you for all of the little moments of humanity we shared in the midst of the graduate school machine { they carried me through. To Katie, Rachel S., Morgan, and Kay { you are the very best of friends. If we really are an average of the people we spend the most time around, I'm glad it's you all. You sharpen me as iron sharpens iron. You are my fiercest supporters and greatest inspirations. Having a roommate in graduate school is no easy task but Krystal, Rebecca, Kristen, and Aimee { you all are champs. Thank you for putting up with countless iv study sessions, for never complaining when school came before keeping the kitchen clean, and for letting me \decorate" the apartment for \prelim season." Thank you to the Young Adult group at Immanuel Baptist Church. If you tired of me requesting prayer for prelims, you never showed it.