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Re: greetings and a question!

  • To: mathgroup at smc.vnet.net
  • Subject: [mg83132] Re: [mg83116] greetings and a question!
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 12 Nov 2007 05:16:11 -0500 (EST)
  • References: <200711110802.DAA04679@smc.vnet.net>

On 11 Nov 2007, at 17:02, dimitris wrote:

> Hello to all of you!
>
> Unfortunately, family, working and research issues
> prevent me from participating to the forum as frequent
> as I used to.
>
> Anyway...
>
> It is my first post since a long time so everybody be patient!
>
> A student of mine came across the following
> equation in a mathematical contest:
>
> In[1]:=
> eq = x^2 + y^2 + (a + b)*x - (a - b)*y + a^2 + b^2 - a - b + 1==0;
>
> (all variables are assumed real)
>
> Of course for Mathematica the solution is rather trivial.
>
> In[1]:=
> $Version
>
> Out[1]=
> "5.2 for Microsoft Windows (June 20, 2005)"
>
> In[2]:=
> eq = x^2 + y^2 + (a + b)*x - (a - b)*y + a^2 + b^2 - a - b + 1==0;
>
> In[3]:=
> Reduce[eq, {a, b, x, y}, Reals]
> ToRules[%]
> eq /. %
>
> Out[3]=
> a == 1 && b == 1 && x == -1 && y == 0
>
> Out[4]=
> {a -> 1, b -> 1, x -> -1, y -> 0}
>
> Out[5]=
> True
>
> Can somebody explain concisely the mathematica concept
> behind this solution? In fact I would be much obliged if somebody
> pointed me out how to obtain the result by hand. Also, by curiosity,
> how Mathematica reaches the result?
>
> Dimitris
>
>


I believe Reduce calls on CylindricalDecomposition:

CylindricalDecomposition[
    x^2 + y^2 + (a + b)*x - (a - b)*y + a^2 + b^2 -
        a - b + 1 == 0, {x, y, a, b}]
x == -1 && y == 0 && a == 1 && b == 1

However, I do not think I want to try to explain here what  
CylindricalDecomposition (an algorithm in real algebraic Geometery  
due to G. Collins, usually known as Cylindrical Algebraic  
Decomposition) does. There are lots of books where you can find it  
explained in detail. A comparatively simple explanation can be found  
in Chapter 8 of B. Mishra, "Algorithmic Algebra", Springer 1993. If  
you are only interested in the general idea without proofs   take a  
look at Mishra's article in the Handbook of Discrete and  
Computational Geometry, edited by Goodman and O'Rourke, CRC Press, 1997.

Andrzej Kozlowski


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