Nonlinear Polynomial Equations Optimization in 2D

I require assistance with solving a system of nonlinear polynomial equations, which can be approached as an unconstrained optimization problem.
Key Requirements:
- Proficiency in handling polynomial equations
- Expertise in using Newton's Method for optimization


Ideal Skills and Experience:
- Strong background in mathematical optimization
- Previous experience with nonlinear polynomial equations
- Proven ability to apply Newton's Method effectively