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## Algebraic systems

AXIOM SESSIONscale=2.5]

```N := NonNegativeInteger

(3)  NonNegativeInteger
Type: Domain
Time: 0 sec

P := Polynomial Integer

(4)  Polynomial Integer
Type: Domain
Time: 0 sec

epsilon := 1/10^21 :: Float

(5)  0.1 E -20
Type: Float
Time: 0 sec

syst(n:N): List P == [x^n+y+z-1, x+y^n+z-1, x+y+z^n-1]

Type: Void
Time: 0 sec

completeExactSolve(syst(1))

(7)  [[z= - %D - %C + 1,y= %C,x= %D]]
Type: List List Equation Fraction Polynomial Integer
Time: 0.03 (IN) = 0.03 sec

completeExactSolve(syst(2))

(8)
[[z= 0,y= 0,x= 1], [z= 0,y= 1,x= 0], [z= 1,y= 0,x= 0],
2
[z= x,y= x,x  + 2x - 1= 0]]
Type: List List Equation Fraction Polynomial Integer
Time: 0.09 (EV) = 0.09 sec

realApproximateSolve(syst(2),[x,y,z],epsilon)

(9)
[[- 2.4142135623 730950488,- 2.4142135623 730950488,- 2.4142135623 730950488]
,
[0.4142135623 730950488,0.4142135623 730950488,0.4142135623 730950488],
[1.0,0.0,0.0], [0.0,1.0,0.0], [0.0,0.0,1.0]]
Type: List List Float
Time: 0.10 (EV) + 0.02 (OT) = 0.12 sec

completeExactSolve(syst(3))

(10)
[[z= - 1,y= 1,x= 1], [z= 0,y= 0,x= 1], [z= 1,y= - 1,x= 1], [z= 0,y= 1,x= 0],
[z= 1,y= 0,x= 0], [z= 1,y= 1,x= - 1],
2          2         3
[z= - y - x,y  + x y + x  - 1= 0,x  - x - 1= 0],
3           3
3                   - x  + 1    - x  + 1  4    2
[z= x,y= x,x  + 2x - 1= 0], [z= --------,y= --------,x  - x  - 2x + 3= 0],
2           2
3          4    3     2
[z= x,y= - x  - x + 1,x  + x  + 2x  + 1= 0],
3               4    3     2
[z= - x  - x + 1,y= x,x  + x  + 2x  + 1= 0]]
Type: List List Equation Fraction Polynomial Integer
Time: 0.01 (IN) + 1.58 (EV) + 0.01 (OT) + 0.13 (GC) = 1.73 sec

realApproximateSolve(syst(3),[x,y,z],epsilon)

(11)
[[0.4533976515 1640376765,0.4533976515 1640376765,0.4533976515 1640376765],
[1.0,1.0,- 1.0], [1.0,0.0,0.0], [0.0,1.0,0.0], [1.0,- 1.0,1.0],
[0.0,0.0,1.0], [- 1.0,1.0,1.0]]
Type: List List Float
Time: 1.56 (EV) + 0.01 (OT) + 0.12 (GC) = 1.69 sec

completeExactSolve(syst(4))

(12)
[[z= 0,y= 0,x= 1], [z= 0,y= 1,x= 0], [z= 1,y= 0,x= 0],
4
[z= x,y= x,x  + 2x - 1= 0],
4               11     8     7     5     4     3
[z= - x  - x + 1,y= x,x   + 2x  - 3x  + 2x  - 4x  + 3x  - 2x + 2= 0],
4          11     8     7     5     4     3
[z= x,y= - x  - x + 1,x   + 2x  - 3x  + 2x  - 4x  + 3x  - 2x + 2= 0],

4            4
- x  + 1     - x  + 1
[z= --------, y= --------,
2            2
11    10    9    8     7     6     4     3    2
x   + x   + x  - x  - 4x  - 4x  + 4x  + 7x  - x  - 5x - 7= 0]
,

4       2     4          8    5     4    2
[z= - y - x  + 1, y  + (x  - 1)y + x  - x  - 2x  + x  + x + 1= 0,
12     9     8     6     5     4     2
x   - 2x  - 3x  + 2x  + 4x  + 3x  - 2x  - 2x - 2= 0]
]
Type: List List Equation Fraction Polynomial Integer
Time: 0.01 (IN) + 9.63 (EV) + 0.03 (OT) + 0.96 (GC) = 10.63 sec

realApproximateSolve(syst(4),[x,y,z],epsilon)

(13)
[[- 0.8755550011 8456161483,- 0.8755550011 8456161483,1.2878847152 284854786]
,
[- 0.8755550011 8456161482,1.2878847152 284854786,- 0.8755550011 8456161482]
,
[1.2878847152 284854786,- 0.8755550011 8456161482,- 0.8755550011 8456161482]
,
[- 1.3953369944 670730188,- 1.3953369944 670730188,- 1.3953369944 670730188]
,
[0.4746266175 6260555033,0.4746266175 6260555033,0.4746266175 6260555033],
[1.0,0.0,0.0], [0.0,1.0,0.0], [0.0,0.0,1.0]]
Type: List List Float
Time: 9.63 (EV) + 0.01 (OT) + 0.83 (GC) = 10.47 sec

-- completeExactSolve(syst(5))    MORE THAN FIVE MUNITES !!!!
```

Next: Conclusions Up: A Second Session Previous: Operations on univariate polynomials
Marc Moreno Maza
2004-04-27