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Families of Thue Inequalities with Transitive AutomorphismsAn, Wenyong January 2014 (has links)
A family of parameterized Thue equations is defined as F_{t,s,...}(X, Y ) = m, m ∈ Z
where F_{t,s,...}(X,Y) is a form in X and Y with degree greater than or equal to 3 and integer coefficients that are parameterized by t, s, . . . ∈ Z. A variety of these families have been studied by different authors.
In this thesis, we study the following families of Thue inequalities
|sx3 −tx2y−(t+3s)xy2 −sy3|≤2t+3s, |sx4 −tx3y−6sx2y2 +txy3 +sy4|≤6t+7s,
|sx6 − 2tx5y − (5t + 15s)x4y2 − 20sx3y3 + 5tx2y4
+(2t + 6s)xy5 + sy6| ≤ 120t + 323s,
where s and t are integers. The forms in question are “simple”, in the sense that the roots of the underlying polynomials can be permuted transitively by automorphisms.
With this nice property and the hypergeometric functions, we construct sequences of good approximations to the roots of the underlying polynomials. We can then prove that under certain conditions on s and t there are upper bounds for the number of integer solutions to the above Thue inequalities.
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The Prouhet-Tarry-Escott problemCaley, Timothy January 2012 (has links)
Given natural numbers n and k, with n>k, the
Prouhet-Tarry-Escott (PTE) problem asks for distinct
subsets of Z, say X={x_1,...,x_n} and
Y={y_1,...,y_n}, such that
x_1^i+...+x_n^i=y_1^i+...+y_n^i\] for i=1,...,k. Many
partial solutions to this problem were found in the late
19th century and early 20th century.
When k=n-1, we call a solution X=(n-1)Y ideal. This is
considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves,
and computational techniques.
This thesis focuses on the ideal case. We extend the framework of the problem to number fields,
and prove generalizations of results from the literature. This information is used along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers.
We also extend a computation from the literature and find new lower bounds for the constant C_n associated to ideal PTE solutions. Further, we present a new algorithm that determines whether an ideal PTE solution with a particular constant exists. This algorithm improves the upper bounds for C_n and in fact, completely determines the value of C_6.
We also examine the connection between elliptic curves and ideal PTE solutions. We use quadratic twists of curves that appear in the literature to find ideal PTE solutions over number fields.
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The Prouhet-Tarry-Escott problemCaley, Timothy January 2012 (has links)
Given natural numbers n and k, with n>k, the
Prouhet-Tarry-Escott (PTE) problem asks for distinct
subsets of Z, say X={x_1,...,x_n} and
Y={y_1,...,y_n}, such that
x_1^i+...+x_n^i=y_1^i+...+y_n^i\] for i=1,...,k. Many
partial solutions to this problem were found in the late
19th century and early 20th century.
When k=n-1, we call a solution X=(n-1)Y ideal. This is
considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves,
and computational techniques.
This thesis focuses on the ideal case. We extend the framework of the problem to number fields,
and prove generalizations of results from the literature. This information is used along with computational techniques to find ideal solutions to the PTE problem in the Gaussian integers.
We also extend a computation from the literature and find new lower bounds for the constant C_n associated to ideal PTE solutions. Further, we present a new algorithm that determines whether an ideal PTE solution with a particular constant exists. This algorithm improves the upper bounds for C_n and in fact, completely determines the value of C_6.
We also examine the connection between elliptic curves and ideal PTE solutions. We use quadratic twists of curves that appear in the literature to find ideal PTE solutions over number fields.
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Entire functions and uniform distribution /Wodzak, Michael A. January 1996 (has links)
Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 87-88). Also available on the Internet.
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Entire functions and uniform distributionWodzak, Michael A. January 1996 (has links)
Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 87-88). Also available on the Internet.
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Dirichlet's Theorem in projective general linear groups and the Absolute Siegel's LemmaPekker, Alexander, January 1900 (has links) (PDF)
Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.
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Diophantine Representation in Thin SequencesBaur, Stefan 21 April 2016 (has links)
No description available.
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The Sum of Two Integer Cubes - RestrictedJonsson, Kenny January 2022 (has links)
We study the size of sets containing sums of two integer cubes such that their representation is unique and also fit between two consecutive integer cubes. We will try to write algorithms that efficiently calculate the size of these sets and also implement these algorithms in PythonTM. Although we will fail to find a non-iterative algorithm, we will find different ways of approximating the size of these sets. We will also find that techniques used in our failed algorithms can be used to calculate the number of integer lattice points inside a circle.
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Diophantine Equations Involving Arithmetic Functions of FactorialsBaczkowski, Daniel M. 12 July 2004 (has links)
No description available.
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Diophantine Equations and Cyclotomic FieldsBartolomé, Boris 26 November 2015 (has links)
No description available.
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