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Dissertation/Thesis Abstract

Roots of Polynomial Congruences
by Welsh, Matthew C, Ph.D., Rutgers The State University of New Jersey, School of Graduate Studies, 2019, 89; 13856719
Abstract (Summary)

In this dissertation we derive and then investigate some consequences of a parametrization of the roots of polynomial congruences. To motivate the later chapters, we begin in chapter two by reviewing known results, presented in our own style, about the roots of the quadratic congruence $\mu^2 \equiv -1 \pmod m$. We also review in chapter two applications of the parametrization to the equidistribution and well-spacing of these roots. In chapter three we generalize this classical parametrization of the roots of a quadratic congruence to the cubic congruence $\mu^3 \equiv 2\pmod m$. Several new phenomena are revealed in our derivation, but a special case of the cubic parametrization is seen to be roughly analogous to the quadratic case. We use this case to prove a spacing property analogous to the well-spacing of the quadratic roots, but unfortunately between the points $\left( \frac{\mu}{m}, \frac{\mu^2}{m}\right)$ instead of the $\frac{\mu}{m}$ themselves. In chapter four we consider this special case for an arbitrary polynomial congruence of any degree, deriving a parametrization for the roots of these congruences. And just as in chapter three, we are able to prove a spacing property for certain points related to the roots. Finally, in chapter five we return to the congruence $\mu^3 \equiv 2 \pmod m$ to explore some of the new phenomena mentioned above with a view towards obtaining equidistribution and well-spacing results. We unfortunately do not prove any concrete results in these directions.

Indexing (document details)
 Advisor: Iwaniec, Henryk Commitee: Miller, Steve, Kontorovich, Alex, Pitt, Nigel School: Rutgers The State University of New Jersey, School of Graduate Studies Department: Mathematics School Location: United States -- New Jersey Source: DAI-B 81/3(E), Dissertation Abstracts International Source Type: DISSERTATION Subjects: Mathematics Keywords: Approximation of roots, Binary cubic forms, Co-type zeta function, Parametrization of roots, Root and ideal correspondence, Spacing and distribution of roots Publication Number: 13856719 ISBN: 9781088326251