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MathGroup Archive 2005

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poly question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58393] poly question
  • From: János <janos.lobb at yale.edu>
  • Date: Thu, 30 Jun 2005 04:37:30 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

I have a polynom  called ftlmat

(Dialog) In[187]:=
ftlmat
(Dialog) Out[187]=
2*a^2*b^2*c + 2*a*b^2*c*d -
   2*b^2*c^2*d + b^2*c^2*d^2 +
   4*a^2*b*c*e - 2*a^2*c^2*e +
   2*a*b*c^2*e + 4*a*b*c*d*e -
   4*a*c^2*d*e - 2*b*c^2*d*e -
   2*c^2*d^2*e + 2*b*c^2*d^2*
    e + 2*a^2*c*e^2 +
   2*a*c^2*e^2 + 2*a*c*d*e^2 +
   c^2*d^2*e^2 + 2*a^2*b*c*f -
   2*a*b^2*d*f + 4*a*b*c*d*f -
   4*b^2*c*d*f - 2*b^2*d^2*f +
   2*b*c*d^2*f + 2*b^2*c*d^2*
    f - 2*a^2*c*e*f +
   4*a*b*c*e*f - 4*a*b*d*e*f -
   4*a*c*d*e*f - 4*b*c*d*e*f -
   4*b*d^2*e*f - 2*c*d^2*e*f +
   4*b*c*d^2*e*f + 4*a*c*e^2*
    f - 2*a*d*e^2*f -
   2*d^2*e^2*f + 2*c*d^2*e^2*
    f + 2*a^2*b*f^2 +
   4*a*b*d*f^2 - 2*b^2*d*f^2 +
   2*b*d^2*f^2 + b^2*d^2*f^2 +
   2*a*b*e*f^2 - 2*b*d*e*f^2 +
   2*b*d^2*e*f^2 +
   2*a*e^2*f^2 + d^2*e^2*f^2

If I do a PolynomialReduce of it the following way, I get:

In[170]:=
PolynomialReduce[ftlmat,
   {a*b*c, a*b*f, a*c*e,
    a*e*f, b*c*d, b*d*f,
    c*d*e, d*e*f}, {a, b, c,
    d, e, f}]
Out[170]=
{{2*a*b + 2*b*d + 4*a*e +
     2*c*e + 4*d*e + 2*a*f +
     4*d*f + 4*e*f,
    -2*b*d - 4*d*e + 2*a*f +
     4*d*f + 2*e*f,
    -2*a*c - 4*c*d + 2*a*e +
     2*c*e + 2*d*e - 2*a*f -
     4*d*f + 4*e*f,
    -2*d*e + 2*e*f,
    -2*b*c + b*c*d - 2*c*e +
     2*c*d*e - 4*b*f + 2*d*f +
     2*b*d*f - 4*e*f +
     4*d*e*f, -2*b*d - 4*d*e -
     2*b*f + 2*d*f + b*d*f -
     2*e*f + 2*d*e*f,
    -2*c*d + c*d*e - 2*d*f +
     2*d*e*f, -2*d*e + d*e*f},
   0}

The result show that the first poly I got - related to a*b*c - has  
all 6 variables, the next has 5 and the rest goes like 5,3,5,4,4,3.   
If I total them it is 35.  My question is what series of polynomials  
should I use in PolymonialReduce to get results which contain the  
least amount of variables each - that is the total of the number of  
variables in each resulting polynom should be minimal, and on the  
same time the number of selected polynoms should be also minimal and  
their construction is "simple" - not necessary the same length as in  
my case - and I should not get any reminder in the result of  
PolynominalReduce.

If I look PolynomialReduce as giving a "vectorization" of the polynom  
regarding to the selected {poly1,poly2,...} base, then the components  
of the result are the "polynomial projections" to the individual base  
polynoms.  I would like to select a base where the resulting  
components have the minimum number of variables per component and I  
want this base to be as simple as possible, that is they also should  
have minimum number of variables in them.  I am sure algebra has some  
theory for it, but my brain is just not recalling it right now.


Any good tip,

János


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