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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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Robert Lewis

Posts: 43
Registered: 7/17/08
Re: Automatic solving of unprepared polynomial equation systems?
Posted: Apr 1, 2013 9:08 AM
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On Sunday, March 31, 2013 10:15:27 AM UTC-4, IV wrote:
> 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


I see that several people have already replied with how to get various systems to solve this.

You have a very simple polynomial system. What you are looking for (the equation that A satisfies) is called a resultant. The best algorithm for finding resultants is the Dixon Resultant. If you google that with my name, "Dixon Resultant Lewis", you will find an expository paper I've written on the subject.

In this case, the resultant for A is A^3 + c2*A^2 - c2*c3*A + c1*c2*A - c1*c2*c3.

For such a small and easy system, many methods will work. However if you ever have a larger system, especially one with parameters (your parameters are c1, c2, c3), you will need Dixon. It is a far superior method to, say, Grobner bases.

Robert H. Lewis
Fordham University
New York





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