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We prove that the bihamiltonian cohomology of a semisimple pencil of Poissonbrackets of hydrodynamic type vanishes for almost all degrees. This implies theexistence of a full dispersive deformation of a semisimple bihamiltonianstructure of hydrodynamic type starting from any infinitesimal deformation.

Deformations of semisimple Poisson pencils of hydrodynamic type are unobstructed