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

Author

Andrew Fletcher Vincent, University of South CarolinaFollow

Date of Award

1-1-2012

Document Type

Campus Access Dissertation

Department

Mathematics

First Advisor

Michael Filaseta

Abstract

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.

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.

Rights

© 2012, Andrew Fletcher Vincent

Recommended Citation

Vincent, A. F.(2012). Classifying Polynomials With Reducible Nonreciprocal Parts and the Factorization of Values of Polynomials. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/1614