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Harmonic functions vanishing on a cone


Let Z be a quadratic harmonic cone in ℝ3. We consider the family \(\mathcal{H}(Z)\) of all harmonic functions vanishing on Z. Is \(\mathcal{H}(Z)\) 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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  1. [1]

    M. L. Agranovsky and Y. Krasnov, Quadratic divisors of harmonic polynomials in Rn, Journal d’Analyse Mathématique 82 (2000), 379–395.

    Article  MATH  Google Scholar 

  2. [2]

    D. H. Armitage, Cones on which entire harmonic functions can vanish, Proceedings of the Royal Irish Academy. Section A. Mathematical and Physical Sciences 92 (1992), 107–110.

    MathSciNet  MATH  Google Scholar 

  3. [3]

    J. Bourgain and Z. Rudnick, On the nodal sets of toral eigenfunctions, Inventiones Mathematicae 185 (2011), 199–237.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience Publishers, New York, 1953.

    MATH  Google Scholar 

  5. [5]

    J. B. Holt, On the irreducibility of Legendre’s polynomials, Proceedings of the London Mathematical Society 2 (1913), 126–132.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    H. Ille, Zur Irreduzibilität der Kugelfunktionen, PhD thesis, Universität Berlin, 1924.

    Google Scholar 

  7. [7]

    A. Logunov and E. Malinnikova, On ratios of harmonic functions, Advances in Mathematics 274 (2015), 241–262.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    A. Logunov and E. Malinnikova, Ratios of harmonic functions with the same zero set, Geometric and Functional Analysis 26 (2016), 909–925.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    D. Mangoubi, A gradient estimate for harmonic functions sharing the same zeros, Electronic Research Announcements in Mathematical Sciences 21 (2014), 62–71.

    MathSciNet  Google Scholar 

  10. [10]

    C. L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abhandlungen der Preußischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse (1929), Nr. 1; reprint in On Some Applications of Diophantine Approximations, Publications of the Scuola Normale Superiore, Vol. 2, Edizioni della Normale, Pisa, 2014, pp. 81–138.

    Google Scholar 

  11. [11]

    T. J. Stieltjes, Letter no: 275 of Oct. 2, 1890, in Correspondance d’Hermite et de Stieltjes. Volume 2, Gauthier-Villars, Paris, 1905, pp. 104–106.

    Google Scholar 

  12. [12]

    G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge; The Macmillan Company, New York, 1944.

    Google Scholar 

  13. [13]

    A. Weller Weiser, Topics in the study of harmonic functions in three dimensions, Master’s thesis, The Hebrew University of Jerusalem, 2014.

    Google Scholar 

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Correspondence to Adi Weller Weiser.

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Mangoubi, D., Weller Weiser, A. Harmonic functions vanishing on a cone. Isr. J. Math. 230, 563–581 (2019).

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