Classifying Polynomials With Reducible Nonreciprocal Parts and the Factorization of Values of Polynomials

<p>Let f(x) be a polynomial with integer coefficients. If either f(x) = xdegff(1/x) or f(x) = -xdegff(1/x), then f(x) is called reciprocal. We refer to the non-reciprocal part of f(x) as the polynomial f(x) removed of its irreducible reciprocal factors. In 1970, Schinzel proved that for a given collection of r+ 1 integers a₀,&hellip,a_r, it is possible to classify the positive integers d₁,&hellip,d_r for which the non-reciprocal part of a₀ + a₁xd₁ + ··· + a_rxd_r is reducible. Specific classification results have been given by Selmer, Tverberg, Ljunggren, Mills, Solan, and Filaseta. In the first chapter of this dissertation, we extend an approach of Filaseta's to obtain classification results for additional sparse polynomials.</p> <p>Let Sbe a finite set of rational primes. For a non-zero integer n, we define [n]_S = &pi_{p in S} |n|_p-1, where |n|_p is the usual p-adic norm of n. In 1984, Stewart applied Baker's theorem to prove non-trivial, computationally effective upper bounds for [n(n + 1)···(n + k)]_S for any integer k > 0. Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for [n(n + 1)]_S and [n2 + 7]_S, where S = {2,3} and S = {2}, respectively. In the second chapter of this dissertation, we extend Stewart's theorem to prove effective upper bounds for [f(n)]_S for an arbitraryf(x) inZ[x] having at least two distinct roots.</p>

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Campus Access Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2012

Author dc:creator

• Vincent, Andrew Fletcher

Contributors dc:contributor

• Michael Filaseta

• Mathematics
• Physical Sciences and Mathematics
• Baker's Theorem
• Computational Number Theory
• Exponential Diophantine Equations
• Number Theory
• p-Adic Analysis
• Polynomials

Statement dc:rights

• © 2012, Andrew Fletcher Vincent

Repository record dc:identifier

https://scholarcommons.sc.edu/etd/1614

OAI identifier oai:identifier

oai:scholarcommons.sc.edu:etd-2615