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We prove that the problem of recognition of the existence of solutions as rational functions for linear homogeneous partial differential or difference equations with polynomial coefficients is algorithmically undecidable.

It is proved that the problem of checking the existence of solutions of linear partial differential equations with polynomial coefficients is algorithmically undecidable. Decidability of the problem of checking the existence of monomial solutions with real and complex exponents is established.

The problem of checking existence of infinitely differentiable solutions of linear partial differential equations with zero boundary conditions is considered. The coefficients of the equations are assumed to be polynomials over Z in independent variables. It is proved that this problem is algorithmically undecidable. This result extends results of our… (More)

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