Approximate a Space Curve near a Singular Point (AsymptoticSolve)

The solution curves of a system of polynomial equations can be approximated by Puiseux series near a singular point of the system. This example shows how to obtain such approximations using AsymptoticSolve.

The solution curve of these algebraic equations has a singular point at .

Find the Puiseux series solutions for positive .

Find the Puiseux series solutions for negative .

Compare the approximation (blue curve) with the exact solution (red dashed curve; intersection curve of the surfaces).