Re: greetings and a question!

*To*: mathgroup at smc.vnet.net*Subject*: [mg83117] Re: greetings and a question!*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>*Date*: Mon, 12 Nov 2007 05:08:25 -0500 (EST)*Organization*: The Open University, Milton Keynes, UK*References*: <fh6df9$4ou$1@smc.vnet.net>

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 Hi Dimitris, Nice to hear from you. According to _The Mathematica Book_, "For polynomial systems Reduce uses cylindrical algebraic decomposition for real domains and Gr=C3=B6bner basis methods for complex domains." See http://documents.wolfram.com/mathematica/book/section-A.9.5 For more insight about the Cylindrical Algebraic Decomposition (CAD) algorithm, you could check the following web sites: http://mathworld.wolfram.com/CylindricalAlgebraicDecomposition.html http://planning.cs.uiuc.edu/node292.html Best regards, -- Jean-Marc