Zeros of harmonic trinomials with complex parameter
DOI:
https://doi.org/10.5269/bspm.66200Resumo
In this paper, we study the number of zeros of harmonic trinomials with complex parameter of the form $p_{a}(z)=z^{n}+a\overline{z}^{k}-1,$ $a\in\mathbb{C},n>k,$ and $gcd(n,k)=1.$ We are interested about two problems, first problem about the location of this roots, seconde problem about the number and the location of roots of the harmonic polynomials.Referências
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[2] M. A. Brilleslyper and Lisbeth E. Schaubroeck, Counting interior roots of trinomials, Math. Mag. 91 (2018), no. 2, 142–150, DOI 10.1080/0025570X.2017.1420332. MR3777918
[3] D. Bshouty, W. Hengartner, and Tiferet Suez, The exact bound on the number of zeros of harmonic polynomials, J. Anal.Math. 67 (1995), 207–218, DOI 10.1007/BF02787790. MR1383494.
[4] D. Khavinson and Grzegorz Swiatek, On the number of zeros of certain harmonic polynomials,Proc. Amer. Math. Soc. 131 (2003), no. 2, 409–414, DOI 10.1090/S0002-9939-02- 06476-6. MR1933331
[5] M. M. Johnson, Book Review: Analytic Geometry by Charles H. Lehmann, Natl. Math. Mag. 17 (1943), no. 6, 280. MR1570119.
[6] O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas; Die
Grundlehren der mathematischen Wissenschaften, Band 126. MR0344463
[7] C. Harrat, Zeros of Harmonic Polynomials with Complex Coefficients, Colloquium Mathematicum, Vol. 168 (2022), No. 1, DOI: 10.4064/cm7874-4-2021 .
[8] H. Lewy, On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), no. 10, 689–692, DOI 10.1090/S0002-9904-1936-06397-4.
MR1563404
[9] A. Melman, Geometry of trinomials, Pacific J. Math. 259 (2012), no. 1, 141–159, DOI 10.2140/pjm.2012.259.141. MR2988487
[10] R. Peretz and J. Schmid, Proceedings of the Ashkelon Workshop on Complex Function Theory, (1996), 203–208, Bar-Ilan Univ., Ramat Gan, 1997.
[11] T. Sheil-Small in Tagesbericht,Mathematisches Forsch. Inst. Oberwolfach, Funktionentheorie, 16-22.2.1992, 19.
[12] A. S. Wilmshurst, The valence of harmonic polynomials, Proc. Amer. Math. Soc. 126 (1998), no. 7, 2077–2081, DOI 10.1090/S0002-9939-98-04315-9. MR1443416.
[13] M. B. Yper, J. Brooks, M. Dorff, R. Howell, and L. Schaubroeck, Zeros of a One-Parameter Family of Harmonic Trinomials, Procceding of the American Mathematical Society, Series B Volume 7, Pages 82–90 (June 17, 2020)
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2025-09-23
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