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

TL;DR

This paper explores geometric properties of degree-three complex multi-affine polynomials with distinct zeros, establishing birational equivalence to a specific variety and examining the zeros' geometric relations.

Contribution

It demonstrates that the variety defined by such polynomials is birationally equivalent to a simple algebraic variety via a Möbius transformation and explores geometric relations among zeros.

Findings

Variety P(z1,z2,z3)=0 is birationally equivalent to z1z2z3 +1=0

The equivalence is given by a Möbius transformation

Geometric curiosity relating zeros of P(z,z,zk) for k=1,2,3

Abstract

In this paper, we consider complex polynomials of degree three with distinct zeros and their polarization ((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 M\"obius transformation. The second goal of this note is to present a geometrical curiosity relating the zeros of P(z,z,zk) for k = 1,2,3, where (z1,z2,z3) is arbitrary point on the variety P(z1,z2 z3) = 0.