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Title: The determinant method and applications
Author: Reuss, Thomas
ISNI:       0000 0004 5359 1596
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2015
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The thesis is structured into 5 chapters as follows: Chapter 1 is an introduction to the tools and methods we use most frequently. Chapter 2 Pairs of k-free Numbers, consecutive square-full Numbers. In this chapter, we refine the approximate determinant method by Heath-Brown. We present applications to asymptotic formulas for consecutive k-free integers, and more generally for k-free integers represented by r-tuples of linear forms. We also show how the method can be used to derive an upper bound for the number of consecutive square-full integers. Finally, we apply the method to make a statement about the size of the fundamental solution of Pell equations. Chapter 3 Power-Free Values of Polynomials. A conjecture by Erdös states that for any irreducible polynomial f of degree d≥3 with no fixed (d-1)-th power prime divisor, there are infinfinitely many primes p such that f(p) is (d-1)-free. We prove this conjecture and derive the corresponding asymptotic formulas. Chapter 4 Integer Points on Bilinear and Trilinear Equations. In the fourth chapter, we derive upper bounds for the number of integer solutions on bilinear or trilinear forms. Chapter 5 In the fifth chapter, we present a method to count the monomials that occur in the projective determinant method when the method is applied to cubic varieties.
Supervisor: Heath-Brown, D. R. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: determinant method ; power-free ; square-free ; polynomials ; trilinear ; bilinear ; multilinear ; forms