Equal values of binary forms at integral points
Estimates for reduced binary forms.
Fifteen problems in number theory.
Finiteness criteria for decomposable form equations
How to explicitly solve a Thue-Mahler equation
Hyperbolicity and integral points off divisors in subgeneral position in projective algebraic varieties
The purpose of this article is twofold. The first is to find the dimension of the set of integral points off divisors in subgeneral position in a projective algebraic variety , where k is a number field. As consequences, the results of Ru-Wong (1991), Ru (1993), Noguchi-Winkelmann (2003) and Levin (2008) are recovered. The second is to show the complete hyperbolicity of the complement of divisors in subgeneral position in a projective algebraic variety
Index form equations in quintic fields
The problem of determining power integral bases in algebraic number fields is equivalent to solving the corresponding index form equations. As is known (cf. Győry [25]), every index form equation can be reduced to an equation system consisting of unit equations in two variables over the normal closure of the original field. However, the unit rank of the normal closure is usually too large for practical use. In a recent paper Győry [27] succeeded in reducing index form equations to systems of unit...
Index form equations in sextic fields: a hard computation
Inhomogeneous norm form equations over function fields
La conjecture de Manin pour certaines surfaces de Châtelet
Following the line of attack of La Bretèche, Browning and Peyre, we prove Manin's conjecture in its strong form conjectured by Peyre for a family of Châtelet surfaces which are defined as minimal proper smooth models of affine surfaces of the form Y² - aZ² = F(X,1), where a = -1, F ∈ ℤ[x₁,x₂] is a polynomial of degree 4 whose factorisation into irreducibles contains two non-proportional linear factors and a quadratic factor which is irreducible over ℚ [i]. This result...
Lösungsanzahl der Gleichung |xd-czyd| = p.
Lösungsanzahl der homogenen Normformengleichung
More cubic surfaces violating the Hasse principle
We generalize L. J. Mordell’s construction of cubic surfaces for which the Hasse principle fails.
Multiplicative dependence in number fields
Multiplicative dependence of shifted algebraic numbers
We show that the set obtained by adding all sufficiently large integers to a fixed quadratic algebraic number is multiplicatively dependent. So also is the set obtained by adding rational numbers to a fixed cubic algebraic number. Similar questions for algebraic numbers of higher degrees are also raised. These are related to the Prouhet-Tarry-Escott type problems and can be applied to the zero-distribution and universality of some zeta-functions.
Multiplicative relations on binary recurrences
Given a binary recurrence , we consider the Diophantine equation with nonnegative integer unknowns , where for 1 ≤ i < j ≤ L, , and K is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.
Multivariate Diophantine equations with many solutions
New solutions to xyz = x+y+z = 1 in integers of quartic fields
Non-monogenity of multiquadratic number fields
Norm form équations.