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Algebraic and Arithmetic Properties of Intersective Polynomials

Mishra, Bhawesh
http://orcid.org/0000-0001-6523-9571
http://rave.ohiolink.edu/etdc/view?acc_num=osu1680820298099428

Abstract Details

2023, Doctor of Philosophy, Ohio State University, Mathematics.
A polynomial, with integer coefficients, is said to be intersective if it has roots modulo every positive integer. This is a special case of a more general notion, involving polynomials with coefficients in more general rings, and is of interest for a variety of reasons. This dissertation deals with the algebraic and the arithmetic aspects of intersective polynomials. In Chapter 2, we obtain two necessary and sufficient conditions, a Galois-theoretic and an arithmetic, for univariate polynomials with coefficients in rings of integers of global fields, to be intersective. Chapter 3 focuses on the study of univariate intersective polynomials, with integer coefficients, that are composed entirely of quadratic factors. In this chapter, we also characterize polynomials with quadratic factors that are minimally intersective, i.e., do not have a proper intersective factor. In Chapter 4, we obtain a characterization of intersectivity of multivariate integral polynomials of the form x1n + x2n+ … + xmn - k in terms of a single congruence equation. In Chapter 5, we obtain necessary and sufficient conditions for intersectivity of polynomials of the form (xq-a1) (xq-a2) ... (xq-an), for any fixed prime q. In the process, we also establish a correspondence between coverings of a vector space over Fq using subspaces and finite subsets of non-zero integers that contain a qth power residue modulo almost every prime. Chapter 6 is concerned with the probability of a randomly chosen integral polynomial being intersective. For instance, we show that the probability of a random univariate polynomial, of a fixed degree, greater than 1, being intersective is zero. On the other hand, the probability that a random reducible univariate polynomial, of a fixed degree greater than 1, is intersective is positive, and converges to a number at least one-half, as n approaches infinity.
Vitaly Bergelson (Advisor)
Alan Loper (Committee Member)
Nimish Shah (Committee Co-Chair)
94 p.
Mathematics
Intersective Polynomials; Prime Power Modulo;

Recommended Citations

Citations

  • Mishra, B. (2023). Algebraic and Arithmetic Properties of Intersective Polynomials [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1680820298099428

    APA Style (7th edition)

  • Mishra, Bhawesh. Algebraic and Arithmetic Properties of Intersective Polynomials. 2023. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1680820298099428.

    MLA Style (8th edition)

  • Mishra, Bhawesh. "Algebraic and Arithmetic Properties of Intersective Polynomials." Doctoral dissertation, Ohio State University, 2023. http://rave.ohiolink.edu/etdc/view?acc_num=osu1680820298099428

    Chicago Manual of Style (17th edition)

osu1680820298099428
114
© 2023, some rights reserved.Creative Commons License Algebraic and Arithmetic Properties of Intersective Polynomials by Bhawesh Mishra is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by The Ohio State University and OhioLINK.