It seems that computer algebra systems like Maple (version 11) and Mathematica (version 7) can not solve all simply solvable equation systems automatically. Let us look e.g. at the equation system [c1=A*B/C, c2=C*A/D, D=c3-A, C=A-B], where c1, c2 and c3 are real or complex constants, A, B, C and D are real or complex variables, and the solutions for the variable A are wanted. The equation system forms a cubic equation in A, and the solutions of the equation system are the solutions of this cubic equation. But the solve command can find neither the cubic equation nor its solutions. I think, the equation system has to be somehow prepared to yield a normal form of equation systems. Is a normal form for polynomial equation systems known?
What is with Buchberger algorithm and Gröbner basis? Maple's (version 11) Groebner[Solve] command could also not find the solutions of the equation system.
We know when we have a system of equations with several variables, then we have to insert the various equations skillfully into the other equations to eliminate single variables. But what is the best way to do that, and how can this be done automatically? Is there an automatic algorithm for the insertion - for the elimination of variables?
Why can computer algebra systems not do that? What have I to do that Maple and Mathematica solve such equation systems automatically?
I have a raw idea for an algorithm. I let determine the variables in each equation. If there is a variable that is only in one equation, I let solve this equation for this variable. If there is a variable that is only in two equations, I let solve this two equations for this variable and link both solutions with an equal sign. But what if after that still one variable is in more than two equations? Which two equations should you choose? Should one try all ways?
Is a mathematical algorithm or a computer algorithm known for such equation systems?