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How to reduce in sagemath a 40th degree equation to a 5th degree equation with a 6th degree equation? All three equations are mod 9

asked 2023-06-05 22:10:22 +0200

Periodic_1_6 gravatar image

How to reduce in sagemath a 40th degree equation to a 5th degree equation with a 6th degree equation? All three equations are mod 9

equation 1

22835963083295358096932575511191922182123945984*x^40 + 456719261665907161938651510223838443642478919680*x^39 - 949119715649463320903760169683914265694526504960*x^38 - 74438103413079337596594904736638418838042150174720*x^37 - 211966311223616472653064458230223650062515432325120*x^36 + 5699423076536127631354075023203536650831985341628416*x^35 + 28973596255008264172921747247784413655190430306795520*x^34 - 271995774077019828088650443324511604201030917638062080*x^33 - 1867697404327342199267901249360368498941431933516644352*x^32 + 9056796651291403018168478696099136247382279833599868928*x^31 + 79160624263907039138284570805006392617165984624088711168*x^30 - 223551997508102686121059656283668963809997686306850734080*x^29 - 2458991015411203904019832329162025635890321461058054651904*x^28 + 4260693753881507670321728714982887826581938225381688475648*x^27 + 59071293374791879685378907558855333817814530563406754217984*x^26 - 65166805833928613316698470531587698330832804559521753071616*x^25 - 1131757178880074916089566167507231334194665050735930352074752*x^24 + 843273738240957848823393929862457096912314552519743678447616*x^23 + 17612018891685450913773925408995240476326126279055993164791808*x^22 - 9965070460907997689837650917987090430658787607861126978600960*x^21 - 224803412228839542185624982039516671368355084084003095634247680*x^20 + 115376363264002396798805411172610298752130761902895261355606016*x^19 + 2360934383698977135600868106052207944902529771066700907860197376*x^18 - 1308815542039402425703070866082728213909461400374939017803726848*x^17 - 20337746511113656868636319478920879710270359595089530799861530624*x^16 + 13558465130296871278633915980747022983007195978102160591142518784*x^15 + 142298688018343424254395657862764431521447510737554686159576104960*x^14 - 118833592326356595084649993445344253636179846197802558968828002304*x^13 - 794065401834625006842542559429023479028263847219436978179087007744*x^12 + 835819614910693852323042241683734160878949541950181684520436105216*x^11 + 3426759016172348767255673212872820400903705074714452879983524184064*x^10 - 4539247226522701303816038738989886579860962357232685201849757728768*x^9 - 10838172582209345115431209424474859020815749759396823997114335887360*x^8 + 18274547905763525031916419971549097309724446260463287119585027817472*x^7 + 22538630042118410739462196918476504623707986472870448264236719144960*x^6 - 51332294008917127137813530189611197158096252035341181816762615726080*x^5 - 22103913759369896814721471064671155443726673363949038033115234369536*x^4 + 89873238137891007192684349732347354457768316526153757296330427465728*x^3 - 13267389606759431082806583489371907647806512881963825978855904528384*x^2 - 73919625356285467948968028725559984622542638023960350194728622704640*x + 42792357763031228784543461920827458618601064112586804151160697595809

equation 2

262144*x^6 + 786432*x^5 - 8306688*x^4
- 17924096*x^3 + 96045792*x^2 + 105138912*x-405220671

equation 3

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Comments

If they are mod 9, why the given polynomial coefficients are so large?

Max Alekseyev gravatar imageMax Alekseyev ( 2023-06-06 06:00:24 +0200 )edit

@Max Alekseyev is there a way in sagemath to reduce mod 9 coefficients? Also

Periodic_1_6 gravatar imagePeriodic_1_6 ( 2023-06-06 10:48:33 +0200 )edit

Define your polynomials upfront over Zmod(9) like in @slelievre's answer. It will automatically take care of reducing all coefficients. Or if you have a polynomial f over integers, you can change its ring to Zmod(9) via f.change_ring(Zmod(9)).

Max Alekseyev gravatar imageMax Alekseyev ( 2023-06-06 15:10:52 +0200 )edit

1 Answer

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answered 2023-06-06 11:46:04 +0200

slelievre gravatar image

Define a polynomial ring over the integers modulo nine.

Then define your polynomials.

They will reduce modulo nine automatically.

Then you can reduce one polynomial modulo the other one.

Step by step below.

Define the polynomial ring and the two polynomials:

sage: R.<x> = Zmod(9)['x']
sage: a = (22835963083295358096932575511191922182123945984*x^40
....:      + 456719261665907161938651510223838443642478919680*x^39
....:      - 949119715649463320903760169683914265694526504960*x^38
....:      - 74438103413079337596594904736638418838042150174720*x^37
....:      - 211966311223616472653064458230223650062515432325120*x^36
....:      + 5699423076536127631354075023203536650831985341628416*x^35
....:      + 28973596255008264172921747247784413655190430306795520*x^34
....:      - 271995774077019828088650443324511604201030917638062080*x^33
....:      - 1867697404327342199267901249360368498941431933516644352*x^32
....:      + 9056796651291403018168478696099136247382279833599868928*x^31
....:      + 79160624263907039138284570805006392617165984624088711168*x^30
....:      - 223551997508102686121059656283668963809997686306850734080*x^29
....:      - 2458991015411203904019832329162025635890321461058054651904*x^28
....:      + 4260693753881507670321728714982887826581938225381688475648*x^27
....:      + 59071293374791879685378907558855333817814530563406754217984*x^26
....:      - 65166805833928613316698470531587698330832804559521753071616*x^25
....:      - 1131757178880074916089566167507231334194665050735930352074752*x^24
....:      + 843273738240957848823393929862457096912314552519743678447616*x^23
....:      + 17612018891685450913773925408995240476326126279055993164791808*x^22
....:      - 9965070460907997689837650917987090430658787607861126978600960*x^21
....:      - 224803412228839542185624982039516671368355084084003095634247680*x^20
....:      + 115376363264002396798805411172610298752130761902895261355606016*x^19
....:      + 2360934383698977135600868106052207944902529771066700907860197376*x^18
....:      - 1308815542039402425703070866082728213909461400374939017803726848*x^17
....:      - 2033774651111365686863631947820879710270359595089530799861530624*x^16
....:      + 13558465130296871278633915980747022983007195978102160591142518784*x^15
....:      + 142298688018343424254395657862764431521447510737554686159576104960*x^14
....:      - 118833592326356595084649993445344253636179846197802558968828002304*x^13
....:      - 794065401834625006842542559429023479028263847219436978179087007744*x^12
....:      + 835819614910693852323042241683734160878949541950181684520436105216*x^11
....:      + 3426759016172348767255673212872820400903705074714452879983524184064*x^10
....:      - 4539247226522701303816038738989886579860962357232685201849757728768*x^9
....:      - 10838172582209345115431209424474859020815749759396823997114335887360*x^8
....:      + 18274547905763525031916419971549097309724446260463287119585027817472*x^7
....:      + 22538630042118410739462196918476504623707986472870448264236719144960*x^6
....:      - 51332294008917127137813530189611197158096252035341181816762615726080*x^5
....:      - 22103913759369896814721471064671155443726673363949038033115234369536*x^4
....:      + 89873238137891007192684349732347354457768316526153757296330427465728*x^3
....:      - 13267389606759431082806583489371907647806512881963825978855904528384*x^2
....:      - 73919625356285467948968028725559984622542638023960350194728622704640*x
....:      + 42792357763031228784543461920827458618601064112586804151160697595809)
sage: b = (262144*x^6 + 786432*x^5 - 8306688*x^4 - 17924096*x^3 + 96045792*x^2
....:      + 105138912*x - 405220671)

Check that they have been reduced modulo nine.

sage: a
7*x^40 + 5*x^39 + x^38 + 3*x^36 + 6*x^35 + 6*x^34 + 3*x^33
+ 8*x^31 + x^30 + 8*x^29 + 3*x^27 + 3*x^24 + x^22 + 5*x^21
+ 7*x^20 + 3*x^17 + 3*x^16 + 3*x^8 + 3*x^7 + 3*x^5 + 3*x^4
sage: b
x^6 + 3*x^5 + 6*x^4 + 7*x^3 + 6*x^2 + 3*x

Compute the remainder of one modulo the other.

sage: a % b
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Comments

@slelievre I did not understand! can't it be done with sagemath?

Periodic_1_6 gravatar imagePeriodic_1_6 ( 2023-06-06 12:40:54 +0200 )edit

This was done with sagemath. What's your concern?

Max Alekseyev gravatar imageMax Alekseyev ( 2023-06-06 15:09:33 +0200 )edit

In my answer,

  • anything after sage: (or after ....: if it spans several lines) is some input you type into Sage
  • anything not starting with those is the output
slelievre gravatar imageslelievre ( 2023-06-06 21:34:41 +0200 )edit

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Asked: 2023-06-05 22:10:22 +0200

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Last updated: Jun 06 '23