Common roots of multivariate polynomials

[psyloe]

I was wondering if it were possible to efficiently solve the common root of 4 polynomials in 4 variables algebraically. I am currently using a gradient descent method, which can find these roots in a couple seconds; however, I am concerned about local minima.

So far I have attempted to use the Caylay-Dixon and Macaualy resultant to solve this problem, but these methods take far more memory to compute than is available. Is there a method that is more efficient than the ones I have tried?

 

[tomprice]

I think that learning about Grobner bases / Buchberger's algorithm will help you.

There are algorithms based on these concepts for solving systems of polynomial equations.