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Letbe an elliptic operator with smooth enough coefficients,  u  a solution to the equation  Lu  = 0 in,an open set. As is well known, ifthen  u  = 0 everywhere in Ω. Let  P  be a polynomial of variables, functionsthe solutions to  Lu  = 0; let. If the coefficients of the operator are (real) analytic thenare analytic andimpliesin Ω. Is the same true for smooth but not analytic coefficients? The question also concerns the polynomials of harmonic quaternion fields. In general, the answer turns out to be negative: the paper provides the relevant counterexamples.

On uniqueness of continuation for polynomials of solutions to second-order elliptic PDE