Solvability of 𝔭-adic diagonal equations
Some aspects of Davenport's work
Some estimates for diagonal equations over p-adic fields
Some observations on the Diophantine equation f(x)f(y) = f(z)²
Let f ∈ ℚ [X] be a polynomial without multiple roots and with deg(f) ≥ 2. We give conditions for f(X) = AX² + BX + C such that the Diophantine equation f(x)f(y) = f(z)² has infinitely many nontrivial integer solutions and prove that this equation has a rational parametric solution for infinitely many irreducible cubic polynomials. Moreover, we consider f(x)f(y) = f(z)² for quartic polynomials.
Some Remarks on Vinogradov's Mean Value Theorem and Tarry's Problem.
Study of rational cubic forms via the circle method [after D.R. Heath-Brown, C. Hooley, and R.C. Vaughan]
Sur la dimension diophantienne des corps p-adiques
Sur un problème de Fermat
Symmetric Diophantine systems
Systems of Homogeneous Equations.
The circle method and pairs of quadratic forms
We give non-trivial upper bounds for the number of integral solutions, of given size, of a system of two quadratic form equations in five variables.
The density of integral points on hypersurfaces of degree at least four
The density of S-integral points in projective space with respect to a quadric
The method of infinite ascent applied on
Each of the Diophantine equations has an infinite number of integral solutions for any positive integer . In this paper, we will show how the method of infinite ascent could be applied to generate these solutions. We will investigate the conditions when , and are pair-wise co-prime. As a side result of this investigation, we will show a method of generating an infinite number of co-prime integral solutions of the Diophantine equation for any co-prime integer pair .
The Tarry-Escott and the "easier" Waring problems.
Über Zerfällungen in paarweis ungleiche Polynomwerte.
Variations on a theme of Runge: effective determination of integral points on certain varieties
We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge’s theorem valid for higher-dimensional varieties, generalizing a uniform version of Runge’s theorem due to Bombieri. We then take up the study of how Runge’s method may be expanded by taking advantage of certain coverings. We prove both a result for arbitrary curves and a more explicit result for superelliptic curves. As an application of our...
Variations sur un thème de Carl Runge
Асимптотическая формула для числа решений одного диофантова уравнения
Асимптотическое разложение для числа решений диафантовой системы Гильберта-Камке с растущим числом слагаемых