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A Bogomolov property for curves modulo algebraic subgroups

Philipp Habegger (2009)

Bulletin de la Société Mathématique de France

Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least 2 . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.

A note on the number of S -Diophantine quadruples

Florian Luca, Volker Ziegler (2014)

Communications in Mathematics

Let ( a 1 , , a m ) be an m -tuple of positive, pairwise distinct integers. If for all 1 i < j m the prime divisors of a i a j + 1 come from the same fixed set S , then we call the m -tuple S -Diophantine. In this note we estimate the number of S -Diophantine quadruples in terms of | S | = r .

Bartz-Marlewski equation with generalized Lucas components

Hayder R. Hashim (2022)

Archivum Mathematicum

Let { U n } = { U n ( P , Q ) } and { V n } = { V n ( P , Q ) } be the Lucas sequences of the first and second kind respectively at the parameters P 1 and Q { - 1 , 1 } . In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation x 2 - 3 x y + y 2 + x = 0 , where ( x , y ) = ( U i , U j ) or ( V i , V j ) with i , j 1 . Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.

Bihomogeneous forms in many variables

Damaris Schindler (2014)

Journal de Théorie des Nombres de Bordeaux

We count integer points on varieties given by bihomogeneous equations using the Hardy-Littlewood method. The main novelty lies in using the structure of bihomogeneous equations to obtain asymptotics in generically fewer variables than would be necessary in using the standard approach for homogeneous varieties. Also, we consider counting functions where not all the variables have to lie in intervals of the same size, which arises as a natural question in the setting of bihomogeneous varieties.

Bounds of general Fréchet classes

Jaroslav Skřivánek (2012)


This paper deals with conditions of compatibility of a system of copulas and with bounds of general Fréchet classes. Algebraic search for the bounds is interpreted as a solution to a linear system of Diophantine equations. Classical analytical specification of the bounds is described.

Density of rational points on cyclic covers of n

Ritabrata Munshi (2009)

Journal de Théorie des Nombres de Bordeaux

We obtain upper bound for the density of rational points on the cyclic covers of n . As n our estimate tends to the conjectural bound of Serre.

Density of solutions to quadratic congruences

Neha Prabhu (2017)

Czechoslovak Mathematical Journal

A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly k prime factors for k > 1 . Building upon a proof by E. M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree n x with k prime factors such that a fixed quadratic equation has exactly 2 k solutions modulo n .

Determinanti polinomiali-esponenziali

Raffaele Marcovecchio (2004)

Bollettino dell'Unione Matematica Italiana

Dati m = 2 o m = 3 numeri algebrici non nulli α = α 1 , , α m tali che α j / α l non è una radice dell'unità per ogni j l , consideriamo una classe di determinanti di Vandermonde generalizzati di ordine quattro G a ; x , al variare di x in Z 4 , connessa con alcuni problemi diofantei. Dimostriamo che il numero delle soluzioni y Z 3 in posizione generica dell'equazione polinomiale-esponenziale disomogenea G a ; 0 , y = 0 non supera una costante esplicita N d dipendente solo da d = [ Q ( α 1 , , α m ) : Q ] .

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