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

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,ar, it is possible to classify the positive integers d1,&hellip,dr for which the non-reciprocal part of a0 + a1xd1 + ··· + arxdr 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\u27s 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 = &pip in S |n|p-1, where |n|p is the usual p-adic norm of n. In 1984, Stewart applied Baker\u27s theorem to prove non-trivial, computationally effective upper bounds for [n(n + 1)···(n + k)]S for any integer k \u3e 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\u27s theorem to prove effective upper bounds for [f(n)]S for an arbitraryf(x) inZ[x] having at least two distinct roots