Solutions to a set of polynomials (Commutative Algebra)

[mtak0114]

Hi

I have a set of nonlinear equations $$f_i(x_1,x_2,x_3...)$$ and I want to find their solutions.

After doing some reading I have come across commutative algebra. So to simplify my problem I have converted my nonlinear equations into a set of polynomials $$p_i(x_1,x_2,x_3...,y_1,y_2...)$$ by introducing new variables (the y's) defined in the $$\Re$$.

How can I find the answer to:

1) whether a solutions exists?

2) if so what is it?

To tackle these two questions I have used mathematica to find solutions with no luck...
I then used mathematica to search for a Groebner Basis which is a new set of polynomials with the same solution space also with no luck...

Is there a way to study the equations analytically to answer at least question 1)
(I could only find theorems for equations defined over the Complex field)...Or if not an answer to 1) some thing I can state about this set of polynomials?

Any help would be greatly appreciated

cheers

M

 

[Hurkyl]

If you can find all complex solutions, can you use that information to find the real solutions?

 

[mtak0114]

I've thought about that...

but my polynomials are equations of motion the solutions shouldn't be complex.
Is their a way to treat equations which are defined in the reals
that transforms them into equations which are complex...

like what you can do with numbers i.e work with 4 real numbers or two complex?

Cheers

M