Diagonales de fractions rationnelles et équations différentielles. (French) Zbl 0534.12018
Groupe Étude Anal. Ultramétrique 10e Année 1982/83, No. 2, Exp. No. 18, 10 p. (1984).
For almost all prime numbers \(p\), the solutions of the differential equations induced by geometry, around a generic point \(t\), converge in the disk \(|x-t|_p<1\). B. Dwork conjectured that this is characteristic.
In order to test the conjecture, the author proves that the diagonal of a rational function satisfies a differential equation with coefficients in \(\mathbb{Q}(x)\). (Dwork has already proved that the diagonal of a rational function is actually involved in Apery’s famous prove of the irrationality of \(\zeta(3)\).) Provided the ring \(\mathbb{Q}(\lambda)[y_1,\ldots,y_{\mu}]\) satisfies a certain hypothesis (H) (this means a particular generation of the ring) the author shows that the conjecture is true. The results are given in terms of the Frobenius functor.
In order to test the conjecture, the author proves that the diagonal of a rational function satisfies a differential equation with coefficients in \(\mathbb{Q}(x)\). (Dwork has already proved that the diagonal of a rational function is actually involved in Apery’s famous prove of the irrationality of \(\zeta(3)\).) Provided the ring \(\mathbb{Q}(\lambda)[y_1,\ldots,y_{\mu}]\) satisfies a certain hypothesis (H) (this means a particular generation of the ring) the author shows that the conjecture is true. The results are given in terms of the Frobenius functor.
Reviewer: Alain Escassut (Aubière)