Dickson Polynomials (monographs And Surveys In Pure And Applied Mathematics)
by Gary L. Mullen /
1993 / English / PDF
4.7 MB Download
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.
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.
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.
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.
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.
