Use this URL to cite or link to this record in EThOS:  http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.667019 
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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Abstract:  
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 kfree Numbers, consecutive squarefull Numbers. In this chapter, we refine the approximate determinant method by HeathBrown. We present applications to asymptotic formulas for consecutive kfree integers, and more generally for kfree integers represented by rtuples of linear forms. We also show how the method can be used to derive an upper bound for the number of consecutive squarefull integers. Finally, we apply the method to make a statement about the size of the fundamental solution of Pell equations. Chapter 3 PowerFree Values of Polynomials. A conjecture by Erdös states that for any irreducible polynomial f of degree d≥3 with no fixed (d1)th power prime divisor, there are infinfinitely many primes p such that f(p) is (d1)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:  HeathBrown, D. R.  Sponsor:  Not available  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.667019  DOI:  Not available  
Keywords:  determinant method ; powerfree ; squarefree ; polynomials ; trilinear ; bilinear ; multilinear ; forms  
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