Cauchy-Mirimanoff Polynomials
Major
Applied Mathematics
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Dr. Kelly Pearson
Presentation Format
Event
Abstract/Description
For an integer n > 2 define Pn (X) = (X + 1)n – Xn – 1. Let En (X) be the remaining factor of Pn (X) in Q [X] after removing X and the cyclotomic factors X + 1 and X2 + X + 1. Then Pn (X) = X(X+1)εn (X2 + X + 1)δn En (X) where for even n εn = δn = 0; for odd n εn = 1 and δn = 0,1,2 according as n = 0, 2, 1 (mod 3). In 1903 Mirimanoff conjectured the irreducibility of En (X) over Q when n is prime. This talk will focus on eliminating any factors of degree six. A characterization of the only possible factors of Pn that are of degree six will be given as well as the primes for which these polynomials are possible factors of En.
Other Affiliations
Science and Mathematics
Cauchy-Mirimanoff Polynomials
For an integer n > 2 define Pn (X) = (X + 1)n – Xn – 1. Let En (X) be the remaining factor of Pn (X) in Q [X] after removing X and the cyclotomic factors X + 1 and X2 + X + 1. Then Pn (X) = X(X+1)εn (X2 + X + 1)δn En (X) where for even n εn = δn = 0; for odd n εn = 1 and δn = 0,1,2 according as n = 0, 2, 1 (mod 3). In 1903 Mirimanoff conjectured the irreducibility of En (X) over Q when n is prime. This talk will focus on eliminating any factors of degree six. A characterization of the only possible factors of Pn that are of degree six will be given as well as the primes for which these polynomials are possible factors of En.