A note on universal Hilbert sets.
Démonstration du théorème de Baker-Feldman via les formes linéaires en deux logarithmes
Nous montrons que l’inégalité de Liouville-Baker-Feldman est une conséquence facile d’une minoration de formes linéaires en deux logarithmes.
The Diophantine equation f(x) = g(y)
Thue equations with composite fields
Index form equations in sextic fields: a hard computation
On the exponential local-global principle
Skolem conjectured that the "power sum" A(n) = λ₁α₁ⁿ + ⋯ + λₘαₘⁿ satisfies a certain local-global principle. We prove this conjecture in the case when the multiplicative group generated by α₁,...,αₘ is of rank 1.
Rational points on
Using the recent isogeny bounds due to Gaudron and Rémond we obtain the triviality of , for and a prime number exceeding . This includes the case of the curves . We then prove, with the help of computer calculations, that the same holds true for in the range , . The combination of those results completes the qualitative study of rational points on undertook in our previous work, with the only exception of .
Catalan’s conjecture
The subject of the talk is the recent work of Mihăilescu, who proved that the equation has no solutions in non-zero integers and odd primes . Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebrated: Before the work of Mihăilescu the most definitive result on Catalan’s problem was due to Tijdeman (1976), who proved that the solutions of Catalan’s equation are bounded by an absolute effective constant.
Quadratic factors of f(x) -g(y)
Catalan without logarithmic forms (after Bugeaud, Hanrot and Mihăilescu)
This is an exposition of the recent work of Bugeaud, Hanrot and Mihăilescu showing that Catalan’s conjecture can be proved without using logarithmic forms and electronic computations.
Quantitative Riemann existence theorem over a number field
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