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Random Polynomials Over Finite Fields RANDOM POLYNOMIALS OVER FINITE FIELDS by Andrew Sharkey A thesis submitted to the Faculty of Science at the University of Glasgow for the degree of . Doctor of Philosophy June 22, 1999 ©Andrew Sharkey 1999 ProQuest Number: 13834241 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 13834241 Published by ProQuest LLC(2019). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346 m Questa tesi e dedicata ad Alessandro Paluello (1975-1997). Preface This thesis is submitted in accordance with the regulations for the degree of Doctor of Philoso­ phy in the University of Glasgow. It is the record of research carried out at Glasgow University between October 1995 and September 1998. No part of it has previously been submitted by me for a degree at any university. I should like to express my gratitude to my supervisor, Prof. R.W.K.Odoni, for his ad­ vice and encouragement throughout the period of research. I would also like to thank Dr. S.D.Gohen for helping out with my supervision, and the E.P.S.R.G. for funding my research. Finally, I would like to thank my family for their support, and ‘the boys’, Brightwell, Ratter, Iain, Rob and Moh, without whom this period of study would not have been nearly so enjoyable. A.Sh. C ontents Preface ii Abstract iv Introduction v 1 Background: Probability Theory 1 1.1 Probability Spaces and M easures .............................................................................. 1 1.2 Random Variables and Independence ........................................................................ 2 1.3 Expectation ................................................................................................................... 3 1.4 Moments, Variance and Covariance ........................................................................... 4 1.5 Characteristic Functions .............................................................................................. 6 1.6 Some Probability D istributions .................... 6 1.7 Convergence Theorems ................................................................................................. 9 1.8 Central Limit Theorems .............................................................................................. 10 1.9 A Note on M om ents .................................................................................................... 12 2 Random Polynomials 13 2 .1 A Simple Model ............................................................................................................. 13 2.2 Several Variables - a Key Lem m a ............................................................................... 15 2.3 The General Model ....................................................................................................... 16 2.4 Random Polynomials versus Random M a p s ............................................................. 18 2.5 Independence ............................................................ 18 3 Inverse and Direct Image Sizes 22 3.1 Definitions ...................................................................................................................... 22 3 .2 The Distribution of ................................................................................................. 23 3.3 The Distribution of C c ................................................................................................. 24 3.4 Applications ................................................................................................................... 25 3.5 Direct Image Size and Classical O ccupancy ............................................................. 26 3.6 The Moments of rj and r f ........................................................................................... 29 iii CONTENTS iv 3.7 The Results for a Polynomial V ector ........................................................................ 32 4 Generalised Inverse-Image Variables 35 4.1 First Generalisation ................................................................................................... 35 4.2 A Simple E xam ple ...................................................................................................... 36 4.3 Second Generalisation .............. 37 4.4 A Simple E x am p le ................................................................................ 38 4.5 Third Generalisation .......................................................................................... 38 4.6 A Simple E x am p le ...................................................................................................... 39 4.7 Moment Calculations ................................................................................................... 39 4.8 Summary ...................................................................................................................... 41 5 Random Character Sums 43 5.1 Characters on ¥q ............................................... 43 5.2 Restrictions on p as q —>■ o o ........................................................................................ 44 5.3 The Asymptotic Behaviour of E* (and hence of E ) ................................................. 45 5.4 First Application .......................................................................................................... 52 5.5 Second Application ................................................................................................ 54 5.6 Third Application ....................................................................................................... 56 5.7 Fourth Application ....................................................................................................... 57 Concluding Remarks 58 Appendices 65 A. A Combinatorial Sieve .................................................................................................... 6 6 B. A Multinomial Expansion .............................................................................................. 6 8 C. Uniform Distributions on the Circle ............................................................................... 69 D. The Modulus of an Isotropic M2-Gaussian Variable .................................................... 70 E. A Note on Bessel F unctions ........................................................................................... 71 F. Table of Characteristic Functions .................................................................................. 72 G. Index of N otation ............................................................................................................. 73 Bibliography 75 A bstract The idea of this thesis is to take some questions about polynomials over finite fields and ‘answer’ them using probability theory; that is, we give the average behaviour of certain properties of polynomials. We tend to deal with multivariate polynomials, so questions about factorisation are not considered. Questions which are considered are ones concerning images and pre-images under a random polynomial mapping, and more generic questions which lead to results on the distributions of certain character sums over finite fields. The methods used are based on those used by Odoni (details in Chapter 2). The probability space from which our random polynomial is chosen is essentially the set of all polynomials up to a given degree d, and we define a random variable associated with this space (for example, the number of zeros of a random polynomial). Once we have enough information about the random variable in question, we obtain asymptotic results about the distribution of this variable by letting both d and the size of the field, g, tend to infinty. The results in this work tend to rely on comparisons between random polynomials (of degree up to d) and random mappings. We therefore do a certain amount of work with random mappings, exploiting nice combinatorial properties which they exhibit, and also using some non-trivial results from the classical theory of random maps. The resulting theorems for random polynomials, when interpreted number-theoretically, are often what one would expect, but every once in a while they cough up a surprise. Introduction Finite Fields The theory of finite fields is a meeting point for several branches of mathematical science, including number theory, combinatorics, computing science, coding theory and cryptography. With their roots firmly embedded in classical number theory, finite fields traditionally been thought of as playing a small part in modern pine mathematics. However, since the dawn of the computer age in the nineteen-seventies, they have enjoyed a massive resurgence in popularity and are now the focus of both pure and applied mathematicians, with applications throughout the computer and telecommunications industries. The study of finite fields can be traced to two brilliant mathematicians - Carl Friedrich Gauss (1777-1855), whose work on the arithmetic of congruences laid down the foundations; and Evariste Galois (1811-1832) who formulated the abstract notions required to construct
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