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'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 = &pip 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.
","abstract_html":"<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 a<sub>0</sub>,&hellip,a<sub>r</sub>, it is possible to classify the positive integers d<sub>1</sub>,&hellip,d<sub>r</sub> for which the non-reciprocal part of a<sub>0</sub> + a<sub>1</sub>xd<sub>1</sub> + ··· + a<sub>r</sub>xd<sub>r</sub> 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]<sub>S</sub> = &pi<sub>p in S</sub> |n|<sub>p</sub>-1, where |n|<sub>p</sub> 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)]<sub>S</sub> for any integer k > 0. Effective upper bounds have also been given by Bennett, Filaseta, and Trifonov for [n(n + 1)]<sub>S</sub> and [n2 + 7]<sub>S</sub>, 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)]<sub>S</sub> for an arbitraryf(x) inZ[x] having at least two distinct roots.</p>","abstract_has_math":false,"creators":["Vincent, Andrew Fletcher"],"institution":null,"degree_name":"Ph.D.","degree_level":"Campus Access Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Michael Filaseta"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-01-01T08:00:00Z","date_published":"2012-01-01T08:00:00Z","updated_at":"2026-07-24T04:38:23Z","subjects":["Mathematics","Physical Sciences and Mathematics","Baker's Theorem","Computational Number Theory","Exponential Diophantine Equations","Number Theory","p-Adic Analysis","Polynomials"],"languages":[],"rights":["© 2012, Andrew Fletcher Vincent"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarcommons.sc.edu/etd/1614","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Michael Filaseta"]},{"key":"dc:creator","label":"Author","values":["Vincent, Andrew Fletcher"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Campus Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Physical Sciences and Mathematics","Baker's Theorem","Computational Number Theory","Exponential Diophantine Equations","Number Theory","p-Adic Analysis","Polynomials"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["© 2012, Andrew Fletcher Vincent"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarcommons.sc.edu/etd/1614"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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'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 = &pip 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.
"]},{"key":"dc:title","label":"Title","values":["Classifying Polynomials With Reducible Nonreciprocal Parts and the Factorization of Values of Polynomials"]}]}],"canonical_facts":{"dc:contributor":["Michael Filaseta"],"dc:creator":["Vincent, Andrew Fletcher"],"dc:description.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,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'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 = &pip 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.
"],"dc:identifier":["https://scholarcommons.sc.edu/etd/1614"],"dc:rights":["© 2012, Andrew Fletcher Vincent"],"dc:subject":["Mathematics","Physical Sciences and Mathematics","Baker's Theorem","Computational Number Theory","Exponential Diophantine Equations","Number Theory","p-Adic Analysis","Polynomials"],"dc:title":["Classifying Polynomials With Reducible Nonreciprocal Parts and the Factorization of Values of Polynomials"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Campus Access Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T04:38:23Z"}