scholarly journals On the equation $a(x^m-1)/(x-1)=b(y^n-1)/(y-1)$: II

1984 ◽  
Vol Volume 7 ◽  
Author(s):  
Tarlok Nath Shorey
Keyword(s):  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
International Audience

International audience Using the theory of linear forms in logarithms we generalize an earlier result with R. Balasubramanian on the equation of the title.

scholarly journals Linear forms in logarithms and exponential Diophantine equations

2020 ◽  
Vol Volume 42 - Special... ◽  
Author(s):  
Rob Tijdeman
Keyword(s):  
Diophantine Equations ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Exponential Diophantine Equations ◽  
International Audience

International audience This paper aims to show two things. Firstly the importance of Alan Baker's work on linear forms in logarithms for the development of the theory of exponential Diophantine equations. Secondly how this theory is the culmination of a series of greater and smaller discoveries.

scholarly journals Effective irrationality measures for quotients of logarithms of rational numbers

2015 ◽  
Vol Volume 38 ◽  
Author(s):  
Yann Bugeaud
Keyword(s):  
Rational Numbers ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
International Audience

International audience We establish uniform irrationality measures for the quotients of the logarithms of two rational numbers which are very close to 1. Our proof is based on a refinement in the theory of linear forms in logarithms which goes back to a paper of Shorey.

NOTE ON FOURIER–STIELTJES COEFFICIENTS OF COIN-TOSSING MEASURES

2020 ◽  
Vol 102 (3) ◽  
pp. 479-489
Author(s):  
XIANG GAO ◽  
SHENGYOU WEN
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Transcendental Number ◽  
Linear Forms In Logarithms ◽  
Multiplicative Semigroup ◽  
Linear Forms ◽  
Coin Tossing

It is known that the Fourier–Stieltjes coefficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the sequence ${\{b^{n}\}}_{n\geq 1}$ where $b$ is multiplicatively independent of 2 and the sequence given by the multiplicative semigroup generated by 3 and 5. The proof is based on elementary combinatorics and lower-bound estimates for linear forms in logarithms from transcendental number theory.

Applications of Linear Forms in Logarithms

2008 ◽  
pp. 411-440
Author(s):  
Yann Bugeaud ◽  
Guillaume Hanrot ◽  
Maurice Mignotte
Keyword(s):  
Linear Forms In Logarithms ◽  
Linear Forms

scholarly journals Mordell’s equation: a classical approach

2015 ◽  
Vol 18 (1) ◽  
pp. 633-646 ◽  
Author(s):  
Michael A. Bennett ◽  
Amir Ghadermarzi
Keyword(s):  
Lower Bounds ◽  
Diophantine Equation ◽  
Classical Approach ◽  
Linear Forms In Logarithms ◽  
Lattice Basis Reduction ◽  
Linear Forms ◽  
Basis Reduction

We solve the Diophantine equation$Y^{2}=X^{3}+k$for all nonzero integers$k$with$|k|\leqslant 10^{7}$. Our approach uses a classical connection between these equations and cubic Thue equations. The latter can be treated algorithmically via lower bounds for linear forms in logarithms in conjunction with lattice-basis reduction.

The hyperelliptic equation over function fields

1983 ◽  
Vol 93 (2) ◽  
pp. 219-230 ◽  
Author(s):  
R. C. Mason
Keyword(s):  
Elliptic Equation ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Integer Coefficients ◽  
Finite Set ◽  
Integer Solutions ◽  
Explicit Bounds

Siegel, in a letter to Mordell of 1925(9), proved that the hyper-elliptic equation y2 = g(x) has only finitely many solutions in integers x and y, where g denotes a square-free polynomial of degree at least three with integer coefficients. Siegel's method reduces the hyperelliptic equation to a finite set of Thue equations f(x, y) = 1, where f denotes a binary form with algebraic coefficients and at least three distinct linear factors; x and y are integral in a fixed algebraic number field. Siegel had already proved that the Thue equations so obtained have only finitely many solutions. However, as is well known, the work of Siegel is ineffective in that it fails to provide bounds on the integer solutions of y2 = g(x). In 1969 Baker (1), using the theory of linear forms in logarithms, employed Siegel's technique to establish explicit bounds on x and y; Baker's result thus reduced the problem of determining all integer solutions of the hyperelliptic equation to a finite amount of computation.

scholarly journals Hierarchical Zonotopal Power Ideals

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Matthias Lenz
Keyword(s):  
Characteristic Polynomial ◽  
Hilbert Series ◽  
Finite Sequence ◽  
Vector Spaces ◽  
Geometric Structures ◽  
Linear Forms ◽  
International Audience ◽  
Powers Of Linear Forms ◽  
Matroid Structure

International audience Zonotopal algebra deals with ideals and vector spaces of polynomials that are related to several combinatorial and geometric structures defined by a finite sequence of vectors. Given such a sequence $X$, an integer $k \geq -1$ and an upper set in the lattice of flats of the matroid defined by $X$, we define and study the associated $\textit{hierarchical zonotopal power ideal}$. This ideal is generated by powers of linear forms. Its Hilbert series depends only on the matroid structure of $X$. It is related to various other matroid invariants, $\textit{e. g.}$ the shelling polynomial and the characteristic polynomial. This work unifies and generalizes results by Ardila-Postnikov on power ideals and by Holtz-Ron and Holtz-Ron-Xu on (hierarchical) zonotopal algebra. We also generalize a result on zonotopal Cox modules due to Sturmfels-Xu. La théorie de l'algèbre "zonotopique'' s'occupe d'idéaux et d'espaces vectoriels de polynômes qui ont un rapport avec plusieurs structures combinatoires et géométriques définies par des suites finies de vecteurs. Étant donné une telle suite $X$, un nombre entier $k \geq -1$ et un ensemble supérieur dans le treillis des plans du matroïde défini par $X$, nous définissons et étudions l'$\textit{idéal hiérarchique zonotopique}$, engendré par des puissances de formes linéaires. Sa série de Hilbert dépend seulement de la structure matroïdale de $X$. Il existe des relations avec d'autres invariants de matroïdes, tels que le polynôme d'épluchage et le polynôme caractéristique. Ce travail unifie et généralise des résultats d'Ardila-Postnikov sur les idéaux de puissances et de Holtz-Ron et Holtz-Ron-Xu sur l'algèbre zonotopique (hiérarchique). Nous généralisons aussi un résultat sur les modules de Cox zonotopiques, dû à Sturmfels-Xu.

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