I have a system of polynomial equations with some equations of order 2 or higher. The coefficients are noisy, so they are random variable with some mean and covariance.
The system has multiple solutions (because it is nonlinear), and I want to solve for the entire solution set.
For example, I could use nonlinear least squares, but this would only give me one of the solutions. I want to estimate the entire solution set. Note that the fact the system is noisy makes it more difficult to estimate each solution.
For example, a single second order polynomial can be solved exactly (using the quadratic equation), but if the coefficients are noisy, the discriminant may be negative.
I can't find any information on this topic. Perhaps, I am not searching the right keywords.
Edit: I edited the question to clarify. Hopefully, this is more clear.
Not possible.
Take the case of a single variable rather than a system of polynomial equations. Small changes in the coefficients lead to large changes in the zeros; this is easily demonstrated using Wilkinson's polynomial.
The situation will not improve in the multivariate polynomial case, and since your coefficients are noisy, your solution set will be entirely garbage.
Sorry man!