Dickson polynomials
著者
書誌事項
Dickson polynomials
（Pitman monographs and surveys in pure and applied mathematics, 65）
Longman Scientific & Technical , J. Wiley, 1993
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注記
Bibliography: p. 186-199
Includes indexes
内容説明・目次
内容説明
Dickson polynomials are closely related with Chebyshev polynomials. They have a variety of algebraic and number theoretic properties and satisfy simple second-order linear differentuation equations and linear recurrences. For suitable parameters they form a commutative sermgroup under composition. Dickson polynominals are of fundamental importance in the theory of permutation polynomials and related topics. In particular they serve as examples of integral polynomials which induce permutations for infinitely many primes. According to 'Schur's conjecture' there are essentially no other examples. Dickson polynonuaLs are also important in cryptology and for pseudoprimality testing. The book provides a comprehensive up-to-date collection of results concerning Dickson polynomials and presents several applications. It also treats generalizations to polynomials in several variables and related rational function like Redei functions. Each of the seven chapters includes exercises and notes. Tables of Dickson polynonuals are given in the Appendix. For most parts of the texi only the basic theory of groups, rings and fields is required.
The proof of 'Schur's Conjecture' is largely self-contained but is based on more advanced results like an estimate for the number of rational points on an absolutely irreducible curve over a finite field. Two important theorems on primitive permutation groups are supplied with complete proofs. The book may serves as a reference text for graduate students or reserachers interested in algebraic or number theoretic aspects of polynomials and for cryptologists.
目次
1. Algebraic foundations: Polynomials, Finite fields, The ring of integers mod m, Galois rings, Exercises, Notes 2. Basic properties of Dickson polynomials: Definition of Dickson polynomials, Algebraic properties, Analytic and arithmetic properties, Several indeterminates, Other related polynomials and functions, Exercises, Notes 3. Finite fields: Permutations, Factorization of Dn(x,a), Value sets, Fixed points, Several indeterminates, Exercises, Notes 4. Integers mod m: Dickson permutations, Fixed points, Several indeterininates, Redei functions, Exercises, Notes 5. Other rings: Galois rings, Infinite algebraic extensions of finite fields, Matrix rings, Exercises, Notes 6. The Schur conjecture and its converse: Prerequisites, Primitive permutation groups, The Galois group of f(x) - t over K(t), Schur's conjecture, Some algebraic number theory, The converse problem, A quantitative refinement, Exercises, Notes 7. Applications: Cryptology, Pseudoprimes, Permutation polynomials and complete mappings, Dickson polynomials and irreducibles, Optimal normal bases, Character sums, Combinatorial applications, Circulant determinanta, exercises, Notes. Appendix: Tables of Dickson polynomials.
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