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

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