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Some geometric properties of the solutions of complex multi-affine polynomials of degree three. (English) Zbl 1307.30008

Summary: In this paper we consider complex polynomials \(p(z)\) of degree three with distinct zeros and their polarization \(P(z_1,z_2,z_3)\) with three complex variables. We show, through elementary means, that the variety \(P(z_1,z_2,z_3)=0\) is birationally equivalent to the variety \(z_1z_2z_3+1=0\). Moreover, the rational map certifying the equivalence is a simple Möbius transformation. The second goal of this note is to present a geometrical curiosity relating the zeros of \(z\mapsto P(z,z,z_k)\) for \(k=1,2,3\), where \((z_1,z_2,z_3)\) is an arbitrary point on the variety \(P(z_1,z_2,z_3)=0\).

MSC:

30C10 Polynomials and rational functions of one complex variable
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
14E05 Rational and birational maps
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References:

[1] Chou, Sh.-Ch., Mechanical Geometry Theorem Proving (1988), D. Reidel Publishing Company · Zbl 0661.14037
[2] Coffman, A.; Frantz, M., Möbius transformations and ellipses, Pi Mu Epsilon J., 12, 339-345 (2007)
[3] Cox, D.; Little, J.; O’Shea, D., Ideals, Varieties, and Algorithms (1992), Springer
[4] Modenov, P. S., Problems in Geometry (1981), Mir
[5] Rahman, Q. I.; Schmeisser, G., Analytic Theory of Polynomials (2002), Oxford Univ. Press Inc.: Oxford Univ. Press Inc. New York · Zbl 1072.30006
[6] Schwerdtfeger, H., Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry (1979), Dover Publications, Inc.: Dover Publications, Inc. New York · Zbl 0484.51007
[7] Sendov, Bl.; Sendov, H., Loci of complex polynomials, part I, Trans. Amer. Math. Soc., 10, 366, 5155-5184 (2014) · Zbl 1298.30005
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