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Topic: Automatic solving of unprepared polynomial equation systems?
Replies: 8   Last Post: Apr 1, 2013 7:50 AM

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Waldek Hebisch

Posts: 267
Registered: 12/8/04
Re: Automatic solving of unprepared polynomial equation systems?
Posted: Mar 31, 2013 10:06 PM
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In sci.math.symbolic 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.

Well, in FriCAS:

(19) -> solve([c1=A*B/C, c2=C*A/D, D=c3-A, C=A-B], [D, B, C, A])

2 2
c2 c3 - A c2 + A c1 - A - c2 c3 + A c2 + A
[D= c3 - A, B= ------------------------, C= -------------------,
c1 c1
2 3
(- c1 - A)c2 c3 + (A c1 + A )c2 + A = 0]
Type: List(List(Equation(Fraction(Polynomial(Integer)))))

The last line is the cubic to determine A, and the other variables
are expressed int term of A.

The only trick is that I put A as _last_ variable. Using natural
order with D last everything is expressed in terms of D.

As other posters mentioned Mathematica and Maple also can solve it.

> Is a mathematical algorithm or a computer algorithm known for such equation
> systems?

Two standard approaches are Groebner bases and triangular systems.

Waldek Hebisch

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