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The Equivalence Problem and Signatures of Algebraic Curves

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The Equivalence Problem and Signatures of Algebraic Curves ABSTRACT RUDDY, MICHAEL GERARD. The Equivalence Problem and Signatures of Algebraic Curves. (Under the direction of Irina Kogan and Cynthia Vinzant). In this thesis we study the application of the differential signature construction to the group equivalence problem for complex algebraic curves under the projective action and its subgroups. Given such an action G, a signature map assigns to a generic algebraic curve an irreducible polynomial, called the signature polynomial, with the property that two curves are G-equivalent if and only if they have the same, up to scaling, signature polynomial. We show that for any action of G, one can construct a pair of rational differential invariants, called classifying invariants, that define a signature map. Given a pair of classifying invariants, elimination algorithms allow one to straightforwardly compute a curve's signature polynomial. In practice, however, these algorithms are computationally intensive, and determination of properties of signature polynomials without their explicit computation is of interest. We derive a formula for the degree of the signature polynomial of a curve in terms of the curve's degree, the size of its symmetry group, and some quantities depending on the choice of classifying invariants. We show that for a given set of classifying invariants and a generic curve of fixed degree, the signature polynomials share the same degree, which is an upper bound. We also show that they share the same monomial support and the same genus. For the projective group and five of its subgroups (the affine, special affine, similarity, Euclidean, and special Euclidean groups), we give sets of classifying invariants and use the degree formula to derive the degree of the signature polynomial for a generic curve as a quadratic function of the curve's degree. © Copyright 2019 by Michael Gerard Ruddy All Rights Reserved The Equivalence Problem and Signatures of Algebraic Curves by Michael Gerard Ruddy A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Mathematics Raleigh, North Carolina 2019 APPROVED BY: Bojko Bakalov Ernest Stitzinger Irina Kogan Cynthia Vinzant Co-chair of Advisory Committee Co-chair of Advisory Committee DEDICATION To my mother, Ana Mar´ıaRuddy Romaguera. ii BIOGRAPHY The author was born in San Juan, Puerto Rico to parents Michael Palmer Ruddy and Ana Mar´ıaRuddy Romaguera. At a young age he moved from Puerto Rico to the small town of Union City, Tennessee. He accidently stumbled into mathematics at the University of Tennessee at Martin, where he graduated in 2014. He then moved to Raleigh, North Carolina to attend graduate school in the Department of Mathematics at North Carolina State University. At the time of this writing, he looks forward to joining the nonlinear algebra research group at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany as a postdoctoral researcher. Outside of teaching, learning, and discussing mathematics, the author enjoys indoor rock climbing, reading, nature, and gardening. iii ACKNOWLEDGEMENTS There are many people I wish to thank for helping me reach this point in my career and complete this work. First, I would like to express my utmost and sincere thanks to my advisors, Irina Kogan and Cynthia Vinzant. Without the incredible amount of patience, advice, insight, and time given, I would not be the mathematician I am today, and this work would not have been possible. I could not have asked for better mentors. I am grateful for the people at NC State who made my time there productive and enjoyable. Thank you to my committee members Bojko Bakalov and Ernert Stitzinger for your guidance. I am also very thankful to Molly Fenn for her advice and mentorship in teaching mathematics. A special thank you to the current and former staff in the mathematics department, Trisha Smith-Clinkscales, Denise Seabrooks, John Craig, and Di Bucklad, who always looked out for the graduate students. I want to thank everyone at the nonlinear algebra semester program at ICERM who helped create such a welcoming and stimulating environment. In particular, insightful conversations with Peter Olver, Justin Chen, Danielle Brake, and Simon Telen directly contributed to ideas and work in this thesis. Thank you to my academic siblings, Faye Pasley Simon and Georgy Scholten, for providing such great support and friendship. I want to also thank everyone who helped me reach graduate school. Thank you to all the professors in the mathematics department at UT Martin, in particular Jason DeVito, for your encouragement. I am especially thankful to my parents for instilling in me the value of education and confidence in myself, as well as providing unconditional love and support. Thank you to my mother for being such a loving and kind role model, and to my father for his advice and companionship in facing adversity. Finally I would like to wholeheartedly thank Anila Yadavalli for her amazing support and infectious humor while we traversed the journey of graduate school together. Your work ethic and drive are a constant inspiration and have greatly motivated me during this process. You help me be the best version of myself. iv TABLE OF CONTENTS List of Tables .......................................... vii List of Figures ......................................... viii Chapter 1 Introduction ................................... 1 Chapter 2 Background .................................... 4 2.1 Equivalence problem and differential signatures................... 5 2.1.1 Equivalence problem for curves and classical moving frames . 10 2.1.2 Differental signature curves.......................... 14 2.2 Algebraic Geometry .................................. 17 2.2.1 Rational maps and their images........................ 20 2.2.2 Algebraic curves and intersection multiplicity................ 23 2.2.3 Image of an algebraic curve under a rational map.............. 28 2.3 Actions and invariants of algebraic groups...................... 33 2.3.1 Algebraic curves ................................ 35 Chapter 3 Differential signatures of algebraic curves ................ 40 3.1 Signatures of algebraic curves............................. 41 3.1.1 Differential invariants and algebraic curves ................. 41 3.1.2 Classifying differential invariants....................... 43 3.1.3 Signature Map ................................. 46 3.2 Properties of signature polynomials.......................... 55 3.2.1 Degree of signature polynomials ....................... 55 3.2.2 Generic monomial support and genus .................... 57 3.2.3 Real signature curves ............................. 59 3.2.4 Examples.................................... 61 Chapter 4 Classical subgroups of the projective general linear group ...... 68 4.1 Classifying invariants.................................. 68 4.1.1 Invariants are Separating ........................... 70 4.2 Generic properties of signature polynomials..................... 78 4.2.1 Generic degree ................................. 78 4.3 Fermat curves...................................... 85 Chapter 5 Further Directions ............................... 88 5.1 Numerical methods................................... 88 5.2 Choice of classifying invariants ............................ 89 5.3 Applications to Invariant Theory........................... 89 5.4 Other generic properties................................ 91 References ............................................ 92 v APPENDICES ......................................... 96 Appendix A Algorithms for Signature Polynomials.................. 97 A.1 Computing the Signature Polynomial...................... 97 A.2 Computing Degree Bounds ...........................100 Appendix B Details for Proofs in Section 4.2.1....................104 B.1 T polynomials and Lemma 4.2.1 ........................104 B.2 Details for Lemma 4.2.3 .............................107 B.3 Details for Lemma 4.2.6 .............................108 Appendix C Fermat Curve Computations.......................112 C.1 Affine .......................................112 C.2 Projective.....................................114 vi LIST OF TABLES (k) Table 4.1 Differential functions in uk = y used construct invariants......... 70 vii LIST OF FIGURES Figure 2.1 Three algebraic curves.............................. 16 Figure 2.2 The Euclidean and special affine signatures respectively of the curves in Figure 2.1..................................... 16 ∗ Figure 3.1 X and SX intersected with V (σa) and V (L) respectively. ......... 62 ∗ ~ Figure 3.2 X and SX intersected with with V (σa~) and V (L) respectively. 62 Figure 3.3 The elliptic curve X and a plot of V (SX )................... 63 Figure 3.4 The real affine points of V (SX ) in bertini real................ 64 Figure 3.5 A closer look around the origin......................... 65 Figure 3.6 An even closer look around the origin..................... 66 viii CHAPTER 1 Introduction In the view of the Felix Klein Erlangen program, geometry is the study of properties invariant under some group of transformations. For example, one can consider Euclidean geometry as the study of properties invariant under rigid motions. From this perspective, the problem of deciding equivalence of objects under a group of transformations G, is akin to asking the question: \when are two object really the `same' with respect to some group G?" Many problems in mathematics and applications can be reformulated in
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