Revision history [back]
 | 1 | initial version |
You don't need to use .variety() for finding minimal polynomial here. Instead, you'd need to eliminate all variables except r by calling J1.elimination_ideal( [x, y, s0, c0, s1, c1] ) and then factoring the generator(s) of the resulted elimination ideal. Among the irreducible factors you'd need those that have a root between 0 and 1, which can be tested with the number_of_roots_in_interval() method.
| | 2 | No.2 Revision |
You don't need to use .variety() for finding minimal polynomial here. Instead, you'd need to eliminate all variables except r by calling J1.elimination_ideal( [x, y, s0, c0, s1, c1] ) and then factoring the generator(s) of the resulted elimination ideal. Among the irreducible factors you'd need those that have a root between 0 and 1, which can be tested with the number_of_roots_in_interval() method.
UPD. It turns out that there are 2 minimal polynomials each having 4 roots in the interval $(0,1)$. They are:
r^30 + (2545815/1801202a - 2449853/900601)r^28 + (338486/900601a + 603176/900601)r^27 + (-446648257/152201569a + 12043728993/2435225104)r^26 + (1383259681/304403138a - 1226101327/152201569)r^25 + (-2015430969/374650016a + 1914137211/187325008)r^24 + (-190717225521/31657926352a + 324665390559/31657926352)r^23 + (2115188315493437/556419713562752a - 7222812032062617/1112839427125504)r^22 + (94112339836993/69552464195344a - 597799751694513/278209856781376)r^21 + (-253291049328919143/188069863184210176a + 212261477857858291/94034931592105088)r^20 + (-4781584174232475/47017465796052544a + 9833948490455267/47017465796052544)r^19 + (37367570590359697/376139726368420352a - 121023099795610641/752279452736840704)r^18 + (20999651247970581/188069863184210176a - 18900003993155813/94034931592105088)r^17 + (10453083474612541/376139726368420352a - 65279351862120215/1504558905473681408)r^16 + (-44064505165782737/752279452736840704a + 4726239042423265/47017465796052544)r^15 + (16227635441524291/3009117810947362816a - 115369507693548305/12036471243789451264)r^14 + (9663004251123147/752279452736840704a - 33032622309102601/1504558905473681408)r^13 + (-49709256497213141/12036471243789451264a + 84729889946852853/12036471243789451264)r^12 + (-5521741764471403/6018235621894725632a + 595756732310263/376139726368420352)r^11 + (34673497535139875/48145884975157805056a - 7436862138499025/6018235621894725632)r^10 + (-345885411536503/3009117810947362816a + 4709623543156535/24072942487578902528)r^9 + (-4590502012308313/96291769950315610112a + 7923331959568715/96291769950315610112)r^8 + (931535286989029/48145884975157805056a - 798863982924283/24072942487578902528)r^7 + (-72056983925195/48145884975157805056a + 1981603083884441/770334159602524880896)r^6 + (-22084555538089/48145884975157805056a + 75321892689733/96291769950315610112)r^5 + (603181898011/6018235621894725632a - 16561591112283/96291769950315610112)r^4 + (-9971674893/6018235621894725632a + 18538673951/6018235621894725632)r^3 + (-9264649743/6018235621894725632a + 31621338863/12036471243789451264)r^2 + (34988843/188069863184210176a - 973560803/3009117810947362816)r - 689105/94034931592105088*a + 9773497/752279452736840704
and
r^30 + (2545815/1801202a - 2449853/900601)r^28 + (22486/12337a - 19960/12337)r^27 + (-446648257/152201569a + 12043728993/2435225104)r^26 + (-928907615/304403138a + 802016915/152201569)r^25 + (1796288795/4870450208a + 118006079/2435225104)r^24 + (81394169511/411553042576a - 237170434365/411553042576)r^23 + (307425189198909/556419713562752a - 850699208797337/1112839427125504)r^22 + (37573295226625/69552464195344a - 246974801001649/278209856781376)r^21 + (-60239193343637863/188069863184210176a + 45658652734554995/94034931592105088)r^20 + (-549608776161751/3616728138157888a + 1125227431684579/3616728138157888)r^19 + (27371283258215057/376139726368420352a - 104755137145182993/752279452736840704)r^18 + (4560692726127805/188069863184210176a - 3981105851513519/94034931592105088)r^17 + (2072252269222557/376139726368420352a - 6123796422206871/1504558905473681408)r^16 + (-5679025703445661/752279452736840704a + 920083549412993/94034931592105088)r^15 + (1776887757881603/3009117810947362816a - 5606403805021457/12036471243789451264)r^14 + (2194126156104539/752279452736840704a - 7191165002066257/1504558905473681408)r^13 + (-19953398578164437/12036471243789451264a + 32145091694720885/12036471243789451264)r^12 + (-429591257748895/6018235621894725632a + 56305615331019/376139726368420352)r^11 + (18329629484328995/48145884975157805056a - 3895636964469777/6018235621894725632)r^10 + (-65451127131541/752279452736840704a + 3523020195864751/24072942487578902528)r^9 + (-2101099156041305/96291769950315610112a + 3631888764561931/96291769950315610112)r^8 + (37361161246969/3703529613473677312a - 31842123800333/1851764806736838656)r^7 + (-24177646026627/48145884975157805056a + 649334834531225/770334159602524880896)r^6 + (-17869643587017/48145884975157805056a + 60925199706821/96291769950315610112)r^5 + (446110416779/6018235621894725632a - 12203233778267/96291769950315610112)r^4 + (-180612269/6018235621894725632a + 1458046367/6018235621894725632)r^3 + (-9264649743/6018235621894725632a + 31621338863/12036471243789451264)r^2 + (34988843/188069863184210176a - 973560803/3009117810947362816)r - 689105/94034931592105088*a + 9773497/752279452736840704
where $a = \sqrt{3}$.
| | 3 | No.3 Revision |
You don't need to use .variety() for finding minimal polynomial here. Instead, you'd need to eliminate all variables except r by calling J1.elimination_ideal( [x, y, s0, c0, s1, c1] ) and then factoring the generator(s) of the resulted elimination ideal. Among the irreducible factors you'd need those that have a root between 0 and 1, which can be tested with the number_of_roots_in_interval() method.
UPD. It turns out that there are 2 minimal polynomials each having 4 roots in the interval $(0,1)$. They are:$(0,1)$:
r^30 + (2545815/1801202a - 2449853/900601)r^28 + (338486/900601a + 603176/900601)r^27 + (-446648257/152201569a + 12043728993/2435225104)r^26 + (1383259681/304403138a - 1226101327/152201569)r^25 + (-2015430969/374650016a + 1914137211/187325008)r^24 + (-190717225521/31657926352a + 324665390559/31657926352)r^23 + (2115188315493437/556419713562752a - 7222812032062617/1112839427125504)r^22 + (94112339836993/69552464195344a - 597799751694513/278209856781376)r^21 + (-253291049328919143/188069863184210176a + 212261477857858291/94034931592105088)r^20 + (-4781584174232475/47017465796052544a + 9833948490455267/47017465796052544)r^19 + (37367570590359697/376139726368420352a - 121023099795610641/752279452736840704)r^18 + (20999651247970581/188069863184210176a - 18900003993155813/94034931592105088)r^17 + (10453083474612541/376139726368420352a - 65279351862120215/1504558905473681408)r^16 + (-44064505165782737/752279452736840704a + 4726239042423265/47017465796052544)r^15 + (16227635441524291/3009117810947362816a - 115369507693548305/12036471243789451264)r^14 + (9663004251123147/752279452736840704a - 33032622309102601/1504558905473681408)r^13 + (-49709256497213141/12036471243789451264a + 84729889946852853/12036471243789451264)r^12 + (-5521741764471403/6018235621894725632a + 595756732310263/376139726368420352)r^11 + (34673497535139875/48145884975157805056a - 7436862138499025/6018235621894725632)r^10 + (-345885411536503/3009117810947362816a + 4709623543156535/24072942487578902528)r^9 + (-4590502012308313/96291769950315610112a + 7923331959568715/96291769950315610112)r^8 + (931535286989029/48145884975157805056a - 798863982924283/24072942487578902528)r^7 + (-72056983925195/48145884975157805056a + 1981603083884441/770334159602524880896)r^6 + (-22084555538089/48145884975157805056a + 75321892689733/96291769950315610112)r^5 + (603181898011/6018235621894725632a - 16561591112283/96291769950315610112)r^4 + (-9971674893/6018235621894725632a + 18538673951/6018235621894725632)r^3 + (-9264649743/6018235621894725632a + 31621338863/12036471243789451264)r^2 + (34988843/188069863184210176a - 973560803/3009117810947362816)r - 689105/94034931592105088*a + 9773497/752279452736840704
and
r^30 + (2545815/1801202a - 2449853/900601)r^28 + (22486/12337a - 19960/12337)r^27 + (-446648257/152201569a + 12043728993/2435225104)r^26 + (-928907615/304403138a + 802016915/152201569)r^25 + (1796288795/4870450208a + 118006079/2435225104)r^24 + (81394169511/411553042576a - 237170434365/411553042576)r^23 + (307425189198909/556419713562752a - 850699208797337/1112839427125504)r^22 + (37573295226625/69552464195344a - 246974801001649/278209856781376)r^21 + (-60239193343637863/188069863184210176a + 45658652734554995/94034931592105088)r^20 + (-549608776161751/3616728138157888a + 1125227431684579/3616728138157888)r^19 + (27371283258215057/376139726368420352a - 104755137145182993/752279452736840704)r^18 + (4560692726127805/188069863184210176a - 3981105851513519/94034931592105088)r^17 + (2072252269222557/376139726368420352a - 6123796422206871/1504558905473681408)r^16 + (-5679025703445661/752279452736840704a + 920083549412993/94034931592105088)r^15 + (1776887757881603/3009117810947362816a - 5606403805021457/12036471243789451264)r^14 + (2194126156104539/752279452736840704a - 7191165002066257/1504558905473681408)r^13 + (-19953398578164437/12036471243789451264a + 32145091694720885/12036471243789451264)r^12 + (-429591257748895/6018235621894725632a + 56305615331019/376139726368420352)r^11 + (18329629484328995/48145884975157805056a - 3895636964469777/6018235621894725632)r^10 + (-65451127131541/752279452736840704a + 3523020195864751/24072942487578902528)r^9 + (-2101099156041305/96291769950315610112a + 3631888764561931/96291769950315610112)r^8 + (37361161246969/3703529613473677312a - 31842123800333/1851764806736838656)r^7 + (-24177646026627/48145884975157805056a + 649334834531225/770334159602524880896)r^6 + (-17869643587017/48145884975157805056a + 60925199706821/96291769950315610112)r^5 + (446110416779/6018235621894725632a - 12203233778267/96291769950315610112)r^4 + (-180612269/6018235621894725632a + 1458046367/6018235621894725632)r^3 + (-9264649743/6018235621894725632a + 31621338863/12036471243789451264)r^2 + (34988843/188069863184210176a - 973560803/3009117810947362816)r - 689105/94034931592105088*a + 9773497/752279452736840704
where $a = \sqrt{3}$.
| | 4 | No.4 Revision |
You don't need to use .variety() for finding minimal polynomial here. Instead, you'd need to eliminate all variables except r by calling J1.elimination_ideal( [x, y, s0, c0, s1, c1] ) and then factoring the generator(s) of the resulted elimination ideal. Among the irreducible factors you'd need those that have a root between 0 and 1, which can be tested with the number_of_roots_in_interval() method.
UPD. It turns out that there are 2 minimal polynomials each having 4 roots in the interval $(0,1)$:
and
where $a = \sqrt{3}$.
| | 5 | No.5 Revision |
You don't need to use .variety() for finding minimal polynomial here. Instead, you'd need to eliminate all variables except r by calling J1.elimination_ideal( [x, y, s0, c0, s1, c1] ) and then factoring the generator(s) of the resulted elimination ideal. Among the irreducible factors you'd need those that have a root between 0 and 1, which can be tested with the number_of_roots_in_interval() method.
UPD. UPD. It turns out that there using AA is an overkill (and slow!), while QuadraticField(3) is well enough. There are 2 minimal polynomials each having 4 roots in the interval $(0,1)$:
r^30 + (2545815/1801202*a - 2449853/900601)*r^28 + (338486/900601*a + 603176/900601)*r^27 + (-446648257/152201569*a + 12043728993/2435225104)*r^26 + (1383259681/304403138*a - 1226101327/152201569)*r^25 + (-2015430969/374650016*a + 1914137211/187325008)*r^24 + (-190717225521/31657926352*a + 324665390559/31657926352)*r^23 + (2115188315493437/556419713562752*a - 7222812032062617/1112839427125504)*r^22 + (94112339836993/69552464195344*a - 597799751694513/278209856781376)*r^21 + (-253291049328919143/188069863184210176*a + 212261477857858291/94034931592105088)*r^20 + (-4781584174232475/47017465796052544*a + 9833948490455267/47017465796052544)*r^19 + (37367570590359697/376139726368420352*a - 121023099795610641/752279452736840704)*r^18 + (20999651247970581/188069863184210176*a - 18900003993155813/94034931592105088)*r^17 + (10453083474612541/376139726368420352*a - 65279351862120215/1504558905473681408)*r^16 + (-44064505165782737/752279452736840704*a + 4726239042423265/47017465796052544)*r^15 + (16227635441524291/3009117810947362816*a - 115369507693548305/12036471243789451264)*r^14 + (9663004251123147/752279452736840704*a - 33032622309102601/1504558905473681408)*r^13 + (-49709256497213141/12036471243789451264*a + 84729889946852853/12036471243789451264)*r^12 + (-5521741764471403/6018235621894725632*a + 595756732310263/376139726368420352)*r^11 + (34673497535139875/48145884975157805056*a - 7436862138499025/6018235621894725632)*r^10 + (-345885411536503/3009117810947362816*a + 4709623543156535/24072942487578902528)*r^9 + (-4590502012308313/96291769950315610112*a + 7923331959568715/96291769950315610112)*r^8 + (931535286989029/48145884975157805056*a - 798863982924283/24072942487578902528)*r^7 + (-72056983925195/48145884975157805056*a + 1981603083884441/770334159602524880896)*r^6 + (-22084555538089/48145884975157805056*a + 75321892689733/96291769950315610112)*r^5 + (603181898011/6018235621894725632*a - 16561591112283/96291769950315610112)*r^4 + (-9971674893/6018235621894725632*a + 18538673951/6018235621894725632)*r^3 + (-9264649743/6018235621894725632*a + 31621338863/12036471243789451264)*r^2 + (34988843/188069863184210176*a - 973560803/3009117810947362816)*r - 689105/94034931592105088*a + 9773497/752279452736840704
and
r^30 + (2545815/1801202*a - 2449853/900601)*r^28 + (22486/12337*a - 19960/12337)*r^27 + (-446648257/152201569*a + 12043728993/2435225104)*r^26 + (-928907615/304403138*a + 802016915/152201569)*r^25 + (1796288795/4870450208*a + 118006079/2435225104)*r^24 + (81394169511/411553042576*a - 237170434365/411553042576)*r^23 + (307425189198909/556419713562752*a - 850699208797337/1112839427125504)*r^22 + (37573295226625/69552464195344*a - 246974801001649/278209856781376)*r^21 + (-60239193343637863/188069863184210176*a + 45658652734554995/94034931592105088)*r^20 + (-549608776161751/3616728138157888*a + 1125227431684579/3616728138157888)*r^19 + (27371283258215057/376139726368420352*a - 104755137145182993/752279452736840704)*r^18 + (4560692726127805/188069863184210176*a - 3981105851513519/94034931592105088)*r^17 + (2072252269222557/376139726368420352*a - 6123796422206871/1504558905473681408)*r^16 + (-5679025703445661/752279452736840704*a + 920083549412993/94034931592105088)*r^15 + (1776887757881603/3009117810947362816*a - 5606403805021457/12036471243789451264)*r^14 + (2194126156104539/752279452736840704*a - 7191165002066257/1504558905473681408)*r^13 + (-19953398578164437/12036471243789451264*a + 32145091694720885/12036471243789451264)*r^12 + (-429591257748895/6018235621894725632*a + 56305615331019/376139726368420352)*r^11 + (18329629484328995/48145884975157805056*a - 3895636964469777/6018235621894725632)*r^10 + (-65451127131541/752279452736840704*a + 3523020195864751/24072942487578902528)*r^9 + (-2101099156041305/96291769950315610112*a + 3631888764561931/96291769950315610112)*r^8 + (37361161246969/3703529613473677312*a - 31842123800333/1851764806736838656)*r^7 + (-24177646026627/48145884975157805056*a + 649334834531225/770334159602524880896)*r^6 + (-17869643587017/48145884975157805056*a + 60925199706821/96291769950315610112)*r^5 + (446110416779/6018235621894725632*a - 12203233778267/96291769950315610112)*r^4 + (-180612269/6018235621894725632*a + 1458046367/6018235621894725632)*r^3 + (-9264649743/6018235621894725632*a + 31621338863/12036471243789451264)*r^2 + (34988843/188069863184210176*a - 973560803/3009117810947362816)*r - 689105/94034931592105088*a + 9773497/752279452736840704
where $a = \sqrt{3}$.
