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Topic: Automatic solving of unprepared polynomial equation systems?
Replies: 9   Last Post: Apr 1, 2013 9:08 AM

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IV

Posts: 62
Registered: 9/1/11
Automatic solving of unprepared polynomial equation systems?
Posted: Mar 31, 2013 10:15 AM
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Hallo,

I'm not a mathematician. I'm a natural scientist.

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?

Thanks.






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