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In the ring of holomorphic functions at the origin of C^2, we consider theequation uf'_x+vf'_y=wf where f and w are given. We introduce intersectionmultiplicities relative to w and f'_y along the branches of f, and we study thesolutions (u,v) using these valuations. As an application, we construct anexplicit functional equation satisfied by f.

Matrice magique associ\'ee \`a un germe de courbe plane et division par l'id\'eal Jacobien