System of 4 quadratic equations with 4 unknowns

[landau_rules]

Hi all,

I am trying to generalize a geometric concept found on page 7 from Steven Weinberg's "Gravitation and Cosmology" from 2 to 3 dimensions.

At the end I am stuck with the following set of equations:

1: a₁+a₂z₃²+a₃z₃z₄+a₄z₄²=0
2: b₁+b₂z₃²+b₃z₃z₅+b₄z₅²=0
3: c₁+c₂z₃²+c₃z₃z₆+c₄z₆²=0
4: d₁+d₂z₄²+d₃z₄z₅+d₄z₅²=0

Mathematica does not solve this with a simple Solve[{name_of_1, name_of_2, name_of_3, name_of_4},{z₃, z₄, z₅, z₆}]-statement.

I am no geek, but I have heard that some applications in cryptography are based on the fact that it is easy to demonstrate that a solution of that kind of problem (once you have it) is indeed a solution, but that it is hard to derive it.

The geometric interpretation is easy, but i don't succed in solving it.

So what can I do, to find a solution or all solutions?

Best greetings from Europe
A.

 

[landau_rules]

the unknowns are z₃, z₄, z₅, z₆

 

[chingkui]

In general, to solve nonlinear system like this is definitely much more difficult than linear. To find one solution, you can use numerical method such as fixed point method. Finding all solutions are much more complicated. There are a number of methods to do that, from numerical to heavily algebraic:
1) Numerical (homotopy) continuation: start with a set of similar equations with known solutions, gradually deform the equations to the system you want to solve, and tracking the changing solutions. See: http://en.wikipedia.org/wiki/Numerical_continuation
For software, PHCpack by Jan Verschelde (http://www.math.uic.edu/~jan/) is particularly suitable.
2) Resultant or Multi-Resultant: Work by generating new equations from the original system of equations, and under certain right conditions, you can create a square matrix and eliminate all but one variables by computing the determinant of it to get a polynomial in one variable which can be solved numerically. No guarantee this will work in general though, but for your system, it seems this technique could be used. See http://planetmath.org/encyclopedia/Resultant.html
3) Grobner Basis: if you apply this method a certain way, it will systematically eliminate all but one variables to form a polynomial just as in 2). The good thing about this method is that it is guarantee to work no matter what. The bad thing, it is computationally intensive, it can eat up all your RAM easily and you also don't know how long it will take for the algorithm to finish the job. But, for a quadratic system like yours, it shouldn't take too long though. See http://en.wikipedia.org/wiki/Gröbner_basis

 

[landau_rules]

Thank you for the competent answer. I am seeking an analytical solution, so I'll try Grobner basis approach. I have not found a code available, so I fear that I would have to write one and first understand how the algorithm works.

 

[chingkui]

Coding the Grobner basis algorithm is no easy task. If you have access to symbolic mathematics software such as MAPLE, you could use the algorithm there. There are also free software you can download such as Magma (http://magma.maths.usyd.edu.au/magma/) and Macaulay2 (http://www.math.uiuc.edu/Macaulay2/). I have not use either of them, so I don't know if they are user friendly.