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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