Solve[] for large polynomial systems

• To: mathgroup at smc.vnet.net
• Subject: [mg14727] Solve[] for large polynomial systems
• From: Gianluca Gorni <gorni at dimi.uniud.it>
• Date: Wed, 11 Nov 1998 17:53:33 -0500
• Sender: owner-wri-mathgroup at wolfram.com

```Hello Mathgroupers!

In solving a large system of polynomial equations I have come across
some ouput that looks a litte funny.

eq is a list of 21 equations (appended below) in the 8 variables

var = {b1, b2, b3, b4, b5, b6, b7, b8}

Each equation is homogeneous of degree 3. Let us try to Solve[]:

sol = Solve[ eq == 0, var ]

This gives a list of 123 solutions, with many repetitions (Union[sol]
leaves only 23 distinct solutions). Now some solution rules are strange
(to me, at least):

e.g.  sol[[34,2]] is   (b2^2)/b3 + b6 -> (b2^2 +b3*b6)/b3

This is not in the form variable -> expression. The right-hand side
contains the variables on the left, and the two sides are Equal[]!

Is this just a cosmetic glitch?

I have Mathematica3.01 on PowerMac.

Gianluca Gorni

eq=
{b5*(b3*b5 - b2*b6), b4*(b4*b6 - b3*b7), b5*(b2^2 - b1*b3 + b3*b6 -
b2*b7), b1*b3*b5 - b4*b5^2 - b1*b2*b6 - 3*b3*b5*b6 + 3*b2*b6^2 +
b2*b5*b7, b5*(b2*b5 - b1*b6 + b6^2 - b5*b7), 3*b2*b3*b5 + b1*b4*b5 -
3*b2^2*b6 - 4*b4*b5*b6 - b1*b2*b7 - 2*b3*b5*b7 + 6*b2*b6*b7, b4*(b3^2 -
b2*b4 + b4*b7 - b3*b8), b1*b2^2 - b1^2*b3 - b2*b3*b5 + b1*b4*b5 -
3*b2^2*b6 + 4*b1*b3*b6 - b4*b5*b6 - 3*b3*b6^2 - b1*b2*b7 + 3*b2*b6*b7 +
b2*b5*b8, b1*b2*b5 - b3*b5^2 - b1^2*b6 - 3*b2*b5*b6 + 4*b1*b6^2 -
3*b6^3 + 2*b5*b6*b7 + b5^2*b8, 3*b3^2*b5 + 3*b2*b4*b5 - 3*b2*b3*b6 +
b1*b4*b6 - 3*b4*b6^2 - 3*b2^2*b7 - b1*b3*b7 - 3*b4*b5*b7 + 3*b3*b6*b7 +
3*b2*b7^2 - b3*b5*b8 + b2*b6*b8, 3*b2^3 - 2*b1*b2*b3 - b1^2*b4 -
b3^2*b5 + b2*b4*b5 + 4*b1*b4*b6 - 3*b4*b6^2 - 6*b2^2*b7 + 3*b1*b3*b7 -
b4*b5*b7 - 3*b3*b6*b7 + 3*b2*b7^2 - b1*b2*b8 + b3*b5*b8 + 3*b2*b6*b8,
b4*(b3*b6 - b2*b7 + b7^2 - b6*b8), 3*b2^2*b5 + b1*b3*b5 - 3*b1*b2*b6 -
4*b3*b5*b6 + 3*b2*b6^2 - b1^2*b7 - 5*b2*b5*b7 + 7*b1*b6*b7 - 6*b6^2*b7
+ 2*b5*b7^2 - b1*b5*b8 + 4*b5*b6*b8, 4*b3*b4*b5 + 2*b2*b4*b6 -
6*b2*b3*b7 - 3*b4*b6*b7 + 3*b3*b7^2 - b4*b5*b8 + b2*b7*b8, 3*b2^2*b3 -
b1*b3^2 - 2*b1*b2*b4 + 3*b2*b4*b6 - 3*b2*b3*b7 + 2*b1*b4*b7 -
3*b4*b6*b7 - 2*b2^2*b8 + b3*b6*b8 + 2*b2*b7*b8, b4^2*b5 + 3*b3*b4*b6 -
3*b3^2*b7 - b2*b4*b7 - b4*b6*b8 + b3*b7*b8, 3*b2*b3*b5 - b1*b3*b6 -
2*b1*b2*b7 - 3*b3*b5*b7 + 3*b2*b6*b7 + 2*b1*b7^2 - 3*b6*b7^2 -
2*b2*b5*b8 + b6^2*b8 + 2*b5*b7*b8, 6*b2*b3^2 - 2*b2^2*b4 - 4*b1*b3*b4 +
4*b3*b4*b6 - 3*b3^2*b7 + 5*b2*b4*b7 - 3*b4*b7^2 - 7*b2*b3*b8 + b1*b4*b8
- b4*b6*b8 + 3*b3*b7*b8 + b2*b8^2, 3*b3^3 - 2*b2*b3*b4 - b1*b4^2 +
b4^2*b6 + 3*b3*b4*b7 - 4*b3^2*b8 - b4*b7*b8 + b3*b8^2, 3*b3^2*b5 +
b2*b4*b5 + 3*b2*b3*b6 - b1*b4*b6 + b4*b6^2 - 3*b2^2*b7 - 3*b1*b3*b7 -
b4*b5*b7 + 6*b2*b7^2 - 3*b7^3 - 4*b3*b5*b8 - 3*b2*b6*b8 + b1*b7*b8 +
2*b6*b7*b8 + b5*b8^2, b3*b4*b5 + 3*b3^2*b6 - 3*b2*b3*b7 - b1*b4*b7 +
b4*b6*b7 + 3*b3*b7^2 - b4*b5*b8 - 4*b3*b6*b8 + b2*b7*b8 - b7^2*b8 +
b6*b8^2}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Gianluca Gorni
Universita` di Udine
Dipartimento di Matematica e Informatica via delle Scienze 208
I-33100 Udine UD
Italy

Ph.:(39) 0432-558422    Fax:(39) 0432-558499 mailto:gorni at dimi.uniud.it
http://www.dimi.uniud.it/~gorni/

```

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