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Weierstrass Theorem University of Bath PHD Use of algebraically independent numbers in computation Elsonbaty, Ahmed Award date: 2004 Awarding institution: University of Bath Link to publication Alternative formats If you require this document in an alternative format, please contact: [email protected] General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 04. Oct. 2021 Use of algebraically independent numbers in computation submitted by Ahmed Elsonbaty for the degree of Doctor of Philosophy of the University of Bath 2004 COPYRIGHT Attention is drawn to the fact that copyright of this thesis rests with its author. This copy of the thesis has been supplied on the condition that anyone who consults it is understood to recognise that its copyright rests with its author and that no quotation from the thesis and no information derived from it may be published without the prior written consent of the author. This thesis may be made available for consultation within the University Library and may be photocopied or lent to other libraries for the purposes of consultation. Signature of Author ......................................................................................................... Ahmed Elsonbaty UMI Number: U601777 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. Dissertation Publishing UMI U601777 Published by ProQuest LLC 2013. Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 ;1 ft i- o r\ r ' ■ 33 d u I-.... l ..,Q f h r o . •JS.WJJi/SSfiaESifceaW Dedication Thanks to God with His beautiful names and His high attributes. Without His approval nothing can happen. I thank him for giving me a very good supporting family. My father, God bless him, was the one who encouraged me to read and seek knowledge despite the fact that he did not have the chance to do so. He used to value and appreciate knowledge and knowledgeable people and he gave me the chance to understand and appreciate why the Qura’n or simply the Koran begins the revelation with the verses {Read in the name of thy Lord and Cherisher, who created, created man, out of a leech-like clot. Proclaim and thy Lord is most bountiful, He who taught (the use of) the pen, taught man that which he knew not.} [96:1-5] ( [S:V] here and in the other Koranic quotations denotes the chapter number and the verse or verses from the Koran to be translated into English ). For this and more I dedicate this little work to my beloved father Abd Elgwad Elsonbaty and all my teachers. Acknowledgements I want to thank the university of Ain Shams in Cairo and the Egyptian culture bureau in London for giving me this opportunity to study for my Ph.D. in Math­ ematics in the respectable university of Bath. They funded me and supervised my progress. I am very happy to express my sincere gratitude to my thesis supervisor Dr. Daniel Richardson for his constant encouragement and endless support. He was very patient with me, helped me at very difficult times, gave me confidence and encouraged me to complete my research and write this thesis. I would also like to thank my colleagues at the department of computer science who were very helpful and kind. I am very happy also to thank all of our friends in Bath who helped me and my family and they never let us alone, especially Mrs. Elizabeth Thomas and Mr. Stanley Smith. I am really grateful to you and wish you the best. This is a very nice occasion to express my gratitude and love to my wife Nashwa Elbadawy who supports me in all times with confidence and belief. I am also grateful to our two little lovely daughters Hedaya and Maryam. Simply, you fill our life with happiness and give us hope for future. Lastly, I cannot forget to thank my friends and my big family in Egypt especially my mother. Her care, love and prayers for me, my brothers and sisters make a real difference. My mother, God bless you and bless all those I owe much. iii Abstract This thesis is concerned with the Zero testing problem for numbers and poly­ nomials presented in the form of polynomial terms i.e., trees with operators on the interior nodes and natural numbers and variables on the frontier. We attempt to decide whether or not such a tree represents the zero polynomial by substitut­ ing algebraically independent real numbers for the variables and attempting to decide whether or not the resulting constant is zero. We got a probabilistic zero recognition test for polynomials which is somewhat more expensive computationally than the usual probabilistic method of choosing random integers in a large interval and evaluating, but which depends on the ability to choose a random point in the unit cube and to approximate a polyno­ mial at that point. Counterexamples to the uniformity and the witness conjectures were discovered but the case of testing polynomials is still plausible. The method for finding counterexamples may help to resolve other related conjectures. We introduced family of new conjectures to deal with sets of constants like the Pfaffian constants defined by Pfaffian functions with rational coefficients and some boundary conditions. This thesis also introduces a new effective proof of the famous Lindemann the­ orem in the form which give us many sets of algebraically independent numbers and more than that it shows the lower bound (called transcendence measure) of some constants compared to the hypothetical ones. Extended Abstract This thesis is concerned with the Zero testing problem for numbers and poly­ nomials presented in the form of polynomial terms i.e., trees with operators on the interior nodes and natural numbers and variables on the frontier. Testing a number or a polynomial to check if it is zero or not is a fundamental problem in theoretical pure mathematics and in applications in all fields without exception; where we process statistical data and solve some equations. We attempt to decide whether or not such a tree represents the zero polynomial by substituting algebraically independent real numbers for the variables and at­ tempting to decide whether or not the resulting constant is zero. Prom this we get a probabilistic zero recognition test for polynomials which is somewhat more expensive computationally than the usual probabilistic method of choosing ran­ dom integers in a large interval and evaluating, but which depends on the ability to choose a random point in the unit cube and to approximate a polynomial at that point. We define Exp-Log expressions as expressions built up from the natural num­ bers using field operations, radicals, exponentials and logarithms. Let | V (E ) \ be the value of expression E\ given choice of principal branch. The Uniformity Conjecture (UC) claims that for expressions in expanded form ( i.e., for any exponential subexpression exp(>l) of E, we have | V{A) | < 1), a small multiple of the syntactic length bounds the number of decimal places needed to distinguish the defined number from zero, if it is non zero. The UC is a simple form of the family of the witness conjectures which may be stated as follows: For each positive rational number A T, denote by En the set of all nested radical exponential and logarithmic expressions A such that ifE is any subexpression of A, then AT1 < \V(E)\ < N. The Witness conjectures say that if A is in S ^, then there is a function u;;v(n) so that |V(i4)| < e~UN^ , where n is the length of A. The strong witness conjecture is that we can take ujn{u) = K n , where K depends on N, and the weak witness conjecture is that we can take un(ti) = K n , where K depends on N These conjectures imply a deterministic zero test for polynomials, which should be compared with the deterministic method of reducing to canonical form. One of the results of this research is that after spending some time trying to resolve the conjecture, counterexamples to the uniformity and the witness conjec­ tures were discovered. This opened a new page in this direction for more thoughts and trials of more sophisticated relations between the values and some param­ eter of the representation of a number e.g., syntactic length, height (maximum number), level of nesting or composition of the functions we call depth. It is notable that the community of exact geometric computation (EGC) has a theoretical-practical approach to the zero testing problem and a very interesting going project toward robust computation.
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