Let F (t, u) ≡ F (u) be a formal power series in t with polynomial coefficients in u. Let F 1 , . . . , F k be k formal power series in t, independent of u. Assume all these series are characterized by a polynomial equationWe prove that, under a mild hypothesis on the form of this equation, these k + 1 series are algebraic, and we give a strategy to compute a polynomial equation for each of them. This strategy generalizes the so-called kernel method and quadratic method, which apply, respectively, to equations that are linear and quadratic in F (u). Applications include the solution of numerous map enumeration problems, among which the hardparticle model on general planar maps.
Dans le cas où k 0 = Q et où l'extension G de G (le groupe G n'étant pas nécessairement isomorpheà S 4 ) est non triviale (i.e. G n'est pas produit direct de G par un sous-groupe d'ordre 2), J. Martinet a conjecturé que le plongement G → G n'est possible que lorsque le nombre de classes au sens restreint h + N de N est pair. Quand G est isomorpheà S 4 , le théorème IV.2 démontre cette conjecture dans le cas totalement réel, et même un résultat plus précis : le nombre de classes au sens restreint du sous-corps K de N de degré 2 sur K est pair.Toutefois,à la suite du théorème IV.3, nous donnons des contre-exemples a la parité de h
The class-invariant homomorphism allows one to measure the Galois module structure of extensions obtained by dividing points on abelian varieties. In this paper, we consider the case when the abelian variety is the Jacobian of a Fermat curve. We give examples of torsion points whose associated Galois structure is trivial, as well as points of infinite order whose associated Galois structure is non-trivial.
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