Re: factoring lists in Singular

clicliclic@xxxxxxxxxx wrote:

c.ro...@xxxxxxxxxxxxx wrote:

Finally, a basic interpretational question, as my algebra is strained
here: if I am interested in all intersections of the two polynomials,
v21l and v22l, the resultant above gives me all combinations of b and
v in the affine variety <v21l,v22l>? I then need to see what further
restrictions are imposed by s?


If what you want are the solutions to the single equation v22l = v21l
and not those to the pair of simultaneous equations [v22l = 0, v21l =
0], then your problem happens to be easy. Turning once more to Derive
6.10:
<snip>
" v22l = v21l is equivalent to v22l - v21l = 0, so let's try: "

FACTOR(v22l-v21l,Rational)
3*b*v^2*(1-v)*(s-1)*(b*s+s*(1-2*v)+v-1)^2*(s*(2*v-1)-v)

Actially, this observation allows easy solution to the original
system. Namely, system [v22l = 0, v21l = 0] has the same solutions
as [v21l = 0, v22l - v21l = 0]. Since v22l - v21l factors we
need just to solve all systems of form [v21l = 0, fi = 0] where
fi is factor of v22l - v21l. Using FriCAS it takes almost no
time (0.08s) to compute Groebner bases of ideals genrated by
the pairs [v21l, fi].

(23) -> factors(v21l - v22l)

(23)
[[factor= 3,exponent= 1], [factor= b,exponent= 1],
[factor= s - 1,exponent= 1], [factor= v - 1,exponent= 1],
[factor= v,exponent= 2], [factor= (2s - 1)v + (- b - 1)s + 1,exponent= 2],
[factor= (2s - 1)v - s,exponent= 1]]
Type: List Record(factor: Polynomial Integer,exponent: Integer)
Time: 0 sec
(24) -> lfd := [v.factor for v in factors(v21l - v22l)]

(24) [3,b,s - 1,v - 1,v,(2s - 1)v + (- b - 1)s + 1,(2s - 1)v - s]
Type: List Polynomial Integer
Time: 0 sec
(25) -> [groebner([v21l, lfd.i]) for i in 2..#lfd]

(25)
2 5 2 4 2 3
[[(4s - 4s + 1)v + (- 4s + 6s - 2)v + (s - 2s + 1)v ,b],
2 5 2 4 2 3 3 2
[(2b - 2b - 1)v + (- 4b + 7b)v + (- b - 2b)v + b v ,s - 1],
2 4 2 3 3 2 2
[v - 1,(2b - 2b)s + (- 4b + 4b)s + (b - b + b - 1)s ],
3 4 3 3 3 2
[v,b s - 2b s + b s ],

[
4 3 2 3 4 3 2 2
(b - 4b + 4b )v + (- b + 6b - 8b )v
+
5 4 3 2 7 6 5 4 6
(- 2b + 10b - 18b + 13b )v + (- 864b + 288b + 480b + 96b )s
+
7 6 5 4 3 5
(552b + 1800b - 968b - 1192b - 192b )s
+
7 6 5 4 3 2 4
(- 170b - 962b - 1134b + 1226b + 944b + 96b )s
+
7 6 5 4 3 2 3
(b + 329b + 287b + 315b - 700b - 232b )s
+
6 5 4 3 2 2
(- 6b - 140b + 100b - 108b + 154b )s
+
6 5 4 3 2 5 4 3 2
(- 2b + 13b - 29b + 27b - 9b )s + 2b - 10b + 16b - 9b
,

5 4 3 2 2 4 3 2
(b - 5b + 8b - 4b )v + (2b - 6b + 4b )v
+
7 6 5 4 6
(- 144b + 48b + 80b + 16b )s
+
7 6 5 4 3 5
(32b + 320b - 128b - 192b - 32b )s
+
7 6 5 4 3 2 4
(- 12b - 28b - 244b + 124b + 144b + 16b )s
+
7 6 5 4 3 2 3
(- 4b + 36b - 52b + 108b - 56b - 32b )s
+
7 6 5 4 3 2 2
(- b + 11b - 39b + 61b - 44b + 12b )s
+
6 5 4 3 2 5 4 3
(2b - 10b + 18b - 14b + 4b )s - b + 3b - 2b
,
(2s - 1)v + (- b - 1)s + 1,

6 5 4 3 2 7 6 5 4 6
(b - 6b + 13b - 12b + 4b )v + (- 288b + 96b + 160b + 32b )s
+
7 6 5 4 3 5
(208b + 592b - 336b - 400b - 64b )s
+
7 6 5 4 3 2 4
(- 56b - 376b - 360b + 440b + 320b + 32b )s
+
7 6 5 4 3 2 3
(4b + 100b + 140b + 92b - 256b - 80b )s
+
7 6 5 4 3 2 2
(2b - 14b - 26b + 14b - 32b + 56b )s
+
7 6 5 4 3 2 6 5 4 3 2
(b - 7b + 19b - 25b + 16b - 4b )s - b + 6b - 13b + 12b - 4b
,

7 6 5 4 7 7 6 5 4 3 6
(9b - 3b - 5b - b )s + (- 11b - 17b + 13b + 13b + 2b )s
+
7 6 5 4 3 2 5
(5b + 21b + 6b - 20b - 11b - b )s
+
7 6 5 4 3 2 4
(- b - 9b - 10b + 4b + 13b + 3b )s
+
6 5 4 3 2 3 5 4 3 2 2
(2b + 3b + b - 3b - 3b )s + (- b + b - b + b )s
]
,

[(2s - 1)v - s,

3 2 3 5 3 2 4 3 2 3
(b - 6b + 12b - 8)v + 16b s + (8b - 48b )s + (4b - 24b + 48b)s
+
3 2 2 3 2
(2b - 12b + 24b - 16)s + (b - 6b + 12b - 8)s
,
3 6 2 5 4 3
b s - 3b s + 3b s - s ]
]
Type: List List Polynomial Integer
Time: 0.02 (EV) + 0.01 (OT) + 0.05 (GC) = 0.08 sec


I belive that other systems can do the same.

--
Waldek Hebisch
hebisch@xxxxxxxxxxxxxxxx
.