Let Z be a quadratic harmonic cone in ℝ 3 . We consider the family H(Z) of all harmonic functions vanishing on Z. Is H(Z) finite or infinite dimensional? Some aspects of this question go back to as early as the 19th century. To the best of our knowledge, no nondegenerate quadratic harmonic cone exists for which the answer to this question is known. In this paper we study the right circular harmonic cone and give evidence that the family of harmonic functions vanishing on it is, maybe surprisingly, finite dimensional. We introduce an arithmetic method to handle this question which extends ideas of Holt and Ille and is reminiscent of Hensel’s Lemma.
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Acknowledgments. We are grateful to Charles Fefferman from whose discussions with the first author the work on this topic originated. We thank Zeev Rudnick for his interest in the present work and for helpful comments. We thank Eran Asaf, Nir Avni, Alexander Logunov, Eugenia Malinnikova and Amit Ophir for interesting discussions. This paper is part of the second author’s research towards a Ph.D. dissertation, conducted at the Hebrew University of Jerusalem. The cases of m = 2 and odd m in Theorem 2 together with Theorems 3 and 4 were proved in . We gratefully acknowledge the support of ISF grant no. 753/14.
© 2019, The Hebrew University of Jerusalem.