# Diophantine Approximations and Diophantine Equations

@inproceedings{Schmidt1991DiophantineAA, title={Diophantine Approximations and Diophantine Equations}, author={Wolfgang M. Schmidt}, year={1991} }

Siegel's lemma and heights.- Diophantine approximation.- The thue equation.- S-unit equations and hyperelliptic equations.- Diophantine equations in more than two variables.

## 260 Citations

On Pillai's Diophantine equation.

- Mathematics
- 2006

Let A, B, a, b and c be fixed nonzero integers. We prove several results on the number of solutions to Pillai’s Diophantine equation Aa −Bby = c in positive unknown integers x and y.

TRANSFERENCE THEOREMS FOR DIOPHANTINE APPROXIMATION WITH WEIGHTS

- MathematicsMathematika
- 2020

In this paper we prove transference inequalities for regular and uniform Diophantine exponents in the weighted setting. Our results generalize the corresponding inequalities that exist in the…

Nevanlinna Theory and Diophantine Approximations

- Mathematics
- 2004

In this note, we will introduce some basic problems and progresses in Nevanlinna theory and Diophantine approximations, say, discuss the abc-conjecture and Hall’s conjecture for integers, and prove…

Mumford's Degree of Contact and Diophantine Approximations

- MathematicsCompositio Mathematica
- 2000

The purpose of this note is to present a somewhat unexpected relation between diophantine approximations and the geometric invariant theory. The link is given by Mumford's degree of contact. We show…

Connections Between Transcendence Degree and Diophantine Estimates

- Mathematics
- 2019

We state two conjectures, which together would yield strong results of algebraic independence related to Schanuel’s Conjecture. Partial results on these conjectures are also discussed. 1991…

Diophantine Approximation and Nevanlinna Theory

- Mathematics
- 2011

Beginning with the work of Osgood [65], it has been known that the branch of complex analysis known as Nevanlinna theory (also called value distribution theory) has many similarities with Roth’s…

An Eective Version of Kronecker's Theorem on Simultaneous Diophantine Approximation

- Mathematics
- 2001

Kronecker’s theorem states that if 1, 1, ::: , n are real algebraic numbers, linearly independent over Q, and if 2R n , then for any

Heights, Transcendence, and Linear Independence on Commutative Group Varieties

- Mathematics
- 2003

Of course it is impossible for four lecturers to cover the whole of diophantine approximation and transcendence theory in 24 hours. So each one has to restrict himself to special aspects.

COMBINATORIAL DIOPHANTINE EQUATIONS AND A REFINEMENT OF A THEOREM ON SEPARATED VARIABLES EQUATIONS

- Mathematics
- 2012

We look at Diophantine equations arising from equating classical counting functions such as perfect powers, binomial coefficients and Stirling numbers of the first and second kind. The proofs of the…

Alternative approach for Siegel's lemma

- Mathematics
- 2016

In this article, we present an alternative approach to show a generalization of Siegel's lemma which is an essential tool in Diophantine problems. Our main statement contains the so-called analytic…