Arithmetic properties in two different structures: characterization of polynomials and r-free integers

Name: Veasna Kim

ThaSH: Variables (Mathematics)

ThaSH: Ratio and proportion

Abstract: In this dissertation, there are two different parts, the first part is about a divided- difference characterization of polynomials over a finite field of characteristic two, the second part is about the distribution of r-free integers in Beatty sequences.In the first part, let F be a finite field of characteristic 2 and let n be an integer ≥ 3. For a function ƒ : F → F, if there is a function h : F → F such that the divided difference ƒ[x1,..., xn] on any distinct elements of F satisfies ƒ[x1,..., xn] = h(x 1 + x„), then ƒ is a polynomial of degree at most n over F. This result complements an earlier work of Davies and Rousseau.In the second part, let r ≥ 2 be a fixed integer. A positive integer n is called r-free if in its canonical representation into prime powers each exponent is < r. The integer 1 is considered to be r-free. We consider Qr(x;a, ß), the number of r-free integers lying in a Beatty sequence [an + ß], 1 ≤ n ≤ X, for an irrational a > 1 with bounded partial quotients, and ß Є [0, a). We prove that, as X →∞,Qr(x;a,ß) = X + 0(x(r+1)/2r log3 x), which improves Victorovich’s result in the case of square-free integers. Moreover, we also prove there exist infinitely many consecutive square-free numbers of the forms [an + ß], [an + ß]+ 1, which improves Dimitrov’s result in 2019.

King Mongkut's Institute of Technology Ladkrabang. Central Library

Address: BANGKOK

Email: Lifelong@kmitl.ac.th

Name: Sukrawan Mavecha

Role: Thesis Advisor

Created: 2022

Modified: 2025-03-22

Issued: 2024-02-12

วิทยานิพนธ์/Thesis

application/pdf

eng

DegreeName: Doctor of philosophy

Level: Doctoral Degree

Descipline: Applied Mathematics

Grantor: King Mongkut's Institute of Technology Ladkrabang

©copyrights King Mongkut's Institute of Technology Ladkrabang

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