Dickson Polynomials
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Dickson Polynomials
Dickson polynomials are closely related with Chebyshev polynomials. They have a variety of algebraic and number theoretic properties and satisfy simple second-order linear differential equations and linear recurrences. For suitable parameters they form a commutative semigroup under composition. Dickson polynomials 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 polynomials 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 polynomials are given in the Appendix. For most parts of the text 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 serve as a reference text for graduate students or researchers interested in algebraic or number theoretic aspects of polynomials and for cryptologists.
Topics in Polynomials of One and Several Variables and Their Applications
Author: Themistocles M. Rassias
language: en
Publisher: World Scientific
Release Date: 1993
This volume presents an account of some of the most important work that has been done on various research problems in the theory of polynomials of one and several variables and their applications. It is dedicated to P L Chebyshev, a leading Russian mathematician.
Poincaré Duality Algebras, Macaulay's Dual Systems, and Steenrod Operations
Author: Dagmar M. Meyer
language: en
Publisher: Cambridge University Press
Release Date: 2005-08-18
A monograph demonstrating remarkable and unexpected interdisciplinary connections in the areas of commutative algebra, invariant theory and algebraic topology.