Example 5 (ex. 16 in paper) 

 

> read
 

> read
 

> A := rtable(1 .. 6, 1 .. 6, [[`+`(`/`(1, `*`(`^`(x, 5))), `/`(`*`(3), `*`(x))), 0, `+`(`/`(1, `*`(`^`(x, 4))), `/`(`*`(4), `*`(x))), 0, `+`(`/`(1, `*`(`^`(x, 4))), `/`(`*`(4), `*`(x))), 0], [0, `+`(`/...
 

> t1, M := RealTime(LocalEigenring(A, x, 10)); -1
 

> t1
 

7.025 (5.1)
 

> valuationM := M[2]; 1; CharacPoly(M[1], x); 1
 

 

0
`+`(`*`(`/`(1, 494400000), `*`(`+`(lambda, `-`(_C[6])), `*`(`+`(`-`(`*`(10, `*`(_C[5]))), `*`(20, `*`(lambda)), `*`(4, `*`(_C[4])), `*`(8, `*`(_C[3])), `-`(`*`(5, `*`(_C[6])))), `*`(`+`(`*`(400, `*`(`...
`+`(`*`(`/`(1, 494400000), `*`(`+`(lambda, `-`(_C[6])), `*`(`+`(`-`(`*`(10, `*`(_C[5]))), `*`(20, `*`(lambda)), `*`(4, `*`(_C[4])), `*`(8, `*`(_C[3])), `-`(`*`(5, `*`(_C[6])))), `*`(`+`(`*`(400, `*`(`...
`+`(`*`(`/`(1, 494400000), `*`(`+`(lambda, `-`(_C[6])), `*`(`+`(`-`(`*`(10, `*`(_C[5]))), `*`(20, `*`(lambda)), `*`(4, `*`(_C[4])), `*`(8, `*`(_C[3])), `-`(`*`(5, `*`(_C[6])))), `*`(`+`(`*`(400, `*`(`...		 (5.2)
 

> DecompositionStartingWithPrecisionK(A, M[1], x, 10); 1
 

 

 

 

 

 

 

rtable(1 .. 6, 1 .. 6, [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]], subtype = Matrix)
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
List of subsystems after a first Decomposition in Q((x)):
List of subsystems after a first Decomposition in Q((x)):
List of subsystems after a first Decomposition in Q((x)):
List of subsystems after a first Decomposition in Q((x)):
List of subsystems after a first Decomposition in Q((x)):
List of subsystems after a first Decomposition in Q((x)):
[1, [[`+`(lambda, `/`(629441, 515)), 1], [`+`(lambda, `/`(91, 10)), 1], [`+`(`*`(`^`(lambda, 2)), `*`(`/`(17, 10), `*`(lambda)), `/`(109, 100)), 1], [`+`(lambda, 6), 1], [`+`(lambda, `-`(`/`(41057, 30...
A multiple of the ramification of each sub system:
We do elementary tranformations and apply the ramification when necessary.
[`+`(`-`(`/`(`*`(2), `*`(`^`(t, 3)))), `/`(`*`(8), `*`(`^`(t, 2))), `-`(`/`(`*`(19), `*`(t)))), x = t], [`+`(`-`(`/`(1, `*`(`^`(t, 5)))), `/`(`*`(`/`(3, 2)), `*`(`^`(t, 3))), `/`(`*`(`/`(19, 12)), `*`...
[`+`(`-`(`/`(`*`(2), `*`(`^`(t, 3)))), `/`(`*`(8), `*`(`^`(t, 2))), `-`(`/`(`*`(19), `*`(t)))), x = t], [`+`(`-`(`/`(1, `*`(`^`(t, 5)))), `/`(`*`(`/`(3, 2)), `*`(`^`(t, 3))), `/`(`*`(`/`(19, 12)), `*`...
[`+`(`-`(`/`(`*`(2), `*`(`^`(t, 3)))), `/`(`*`(8), `*`(`^`(t, 2))), `-`(`/`(`*`(19), `*`(t)))), x = t], [`+`(`-`(`/`(1, `*`(`^`(t, 5)))), `/`(`*`(`/`(3, 2)), `*`(`^`(t, 3))), `/`(`*`(`/`(19, 12)), `*`...		 (5.3)
 

> t1, M := RealTime(Eigenring_PositiveValuation(A, x, 10)); -1
 

> t1
 

.538 (5.4)
 

> DecompositionStartingWithPrecisionK(A, M, x, 10); 1
 

 

 

 

 

 

 

rtable(1 .. 6, 1 .. 6, [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]], subtype = Matrix)
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
P the transformation that decompose the Kernel of T
List of subsystems after a first Decomposition in Q((x)):
List of subsystems after a first Decomposition in Q((x)):
List of subsystems after a first Decomposition in Q((x)):
List of subsystems after a first Decomposition in Q((x)):
List of subsystems after a first Decomposition in Q((x)):
List of subsystems after a first Decomposition in Q((x)):
[1, [[`+`(lambda, `-`(`/`(8785243, 976439))), 1], [`+`(lambda, `/`(8103775, 976439)), 1], [`+`(lambda, `/`(8520357, 976439)), 1], [`+`(`*`(`^`(lambda, 2)), `-`(`*`(`/`(121611, 976439), `*`(lambda))), ...
A multiple of the ramification of each sub system:
We do elementary tranformations and apply the ramification when necessary.
[`+`(`/`(1, `*`(`^`(t, 5))), `/`(1, `*`(`^`(t, 4))), `-`(`/`(`*`(`/`(5, 2)), `*`(`^`(t, 3)))), `/`(`*`(`/`(15, 4)), `*`(`^`(t, 2))), `-`(`/`(`*`(`/`(17, 4)), `*`(t)))), x = t], [`+`(`-`(`/`(`*`(2), `*...
[`+`(`/`(1, `*`(`^`(t, 5))), `/`(1, `*`(`^`(t, 4))), `-`(`/`(`*`(`/`(5, 2)), `*`(`^`(t, 3)))), `/`(`*`(`/`(15, 4)), `*`(`^`(t, 2))), `-`(`/`(`*`(`/`(17, 4)), `*`(t)))), x = t], [`+`(`-`(`/`(`*`(2), `*...
[`+`(`/`(1, `*`(`^`(t, 5))), `/`(1, `*`(`^`(t, 4))), `-`(`/`(`*`(`/`(5, 2)), `*`(`^`(t, 3)))), `/`(`*`(`/`(15, 4)), `*`(`^`(t, 2))), `-`(`/`(`*`(`/`(17, 4)), `*`(t)))), x = t], [`+`(`-`(`/`(`*`(2), `*...		 (5.5)