Some geometric properties of the solutions of complex multi-affine polynomials of degree three

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:426

Article

Planiden, Chayne¹  Sendov, Hristo² 

[1] Univ British Columbia, Kelowna, BC V1V 1V7, Canada

[2] Univ Western Ontario, Dept Stat & Actuarial Sci, London, ON N6A 5B7, Canada

关键词: Polynomial;    Multi-affine symmetric polynomial;    Mobius transformation;    Variety;    Birational equivalence;    Polarization;   

DOI  :  10.1016/j.jmaa.2014.12.050

来源: Elsevier

【 摘 要 】

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(1)z(2)z(3) + 1 = 0. Moreover, the rational map certifying the equivalence is a simple Mobius transformation. The second goal of this note is to present a geometrical curiosity relating the zeros of z -> P(z, z, z(k)) for k = 1,2,3, where (z(1), z(2), z(3)) is arbitrary point on the variety P(z(1), z(2), z(3)) = 0. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free