Points entiers et modèles des courbes algébriques.
Points entiers sur les courbes de genre 0
Points entiers sur les courbes hyperelliptiques
Points rationnels et méthode de Chabauty elliptique
La méthode de Chabauty elliptique permet de calculer les points rationnels sur une courbe définie sur un corps de nombres lorsque le théorème de Chabauty ne s’applique pas, c’est à dire lorsque le rang de la jacobienne est supérieur au genre de la courbe. Nous exposons cette méthode et nous la généralisons dans de nouveaux cas en écrivant une version explicite du théorème de préparation de Weierstrass en variables. En particulier nous calculons tous les points rationnels d’une courbe de genre...
Polynômes singuliers à plusieurs variables sur un corps fini et congruences modulo p²
Polynomial cycles in certain rings of rationals
It is shown that the methods established in [HKN3] can be effectively used to study polynomial cycles in certain rings. We shall consider the rings and shall describe polynomial cycles in the case when is either odd or twice a prime.
Polynomial operations: Numerical performance in matrix diophantine equation
Polynomial parametrizations of length 4 Büchi sequences
Polynomial squares of the form
Polynomials dividing infinitely many quadrinomials or quintinomials
Positive solutions of the Diophantine equation.
Power integral bases in orders of composite fields.
Power integral bases in the family of simplest quartic fields.
Power values of sums of products of consecutive integers
We investigate power values of sums of products of consecutive integers. We give general finiteness results, and also give all solutions when the number of terms in the sum considered is at most ten.
Power-free values, large deviations, and integer points on irrational curves
Let be a polynomial of degree without roots of multiplicity or . Erdős conjectured that, if satisfies the necessary local conditions, then is free of th powers for infinitely many primes . This is proved here for all with sufficiently high entropy.The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov’s theorem from the theory of large deviations.
Poznámka o počte riešení optickej rovnice
Poznámka o počte riešení rovnice v prirodzených číslach
p-primary parts of unit traces and the p-adic regulator
Prime divisors of some shifted products.
Prime Pythagorean triangles.