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Some geometric properties of the solutions of complex multiaffine polynomials of degree three

Journal Article


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Abstract


  • In this paper we consider complex polynomials p(z) of degree three with distinct zeros and their polarization P(z1, z2, z3) with three complex variables. We show, through elementary means, that the variety P(z1, z2, z3)=0 is birationally equivalent to the variety z1z2z3+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&P(z, z, zk) for k=1, 2, 3, where (z1, z2, z3) is arbitrary point on the variety P(z1, z2, z3)=0.

Publication Date


  • 2015

Geographic Focus


Citation


  • Planiden, C. & Sendov, H. (2015). Some geometric properties of the solutions of complex multiaffine polynomials of degree three. Journal of Mathematical Analysis and Applications, 426 (1), 312-329.

Scopus Eid


  • 2-s2.0-84922939216

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2115&context=eispapers1

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/1113

Number Of Pages


  • 17

Start Page


  • 312

End Page


  • 329

Volume


  • 426

Issue


  • 1

Place Of Publication


  • United States

Abstract


  • In this paper we consider complex polynomials p(z) of degree three with distinct zeros and their polarization P(z1, z2, z3) with three complex variables. We show, through elementary means, that the variety P(z1, z2, z3)=0 is birationally equivalent to the variety z1z2z3+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&P(z, z, zk) for k=1, 2, 3, where (z1, z2, z3) is arbitrary point on the variety P(z1, z2, z3)=0.

Publication Date


  • 2015

Geographic Focus


Citation


  • Planiden, C. & Sendov, H. (2015). Some geometric properties of the solutions of complex multiaffine polynomials of degree three. Journal of Mathematical Analysis and Applications, 426 (1), 312-329.

Scopus Eid


  • 2-s2.0-84922939216

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2115&context=eispapers1

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/1113

Number Of Pages


  • 17

Start Page


  • 312

End Page


  • 329

Volume


  • 426

Issue


  • 1

Place Of Publication


  • United States