SageMathで数学の力を上げます PyCon mini Hiroshima 2019 / Enlarge your ability of mathematics with SageMath

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SageMathͰ ਺ֶͷ ྗΛ ্͛·͢ Enlarge your ability of mathematics with SageMath ๺㝳೗๏ PyCon mini Hiroshima 2019 @޿ౡࢢཱେֶαςϥΠτΩϟϯύε @2019೥10݄12೔(౔)

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͜ͷࢿྉ͸Ξοϓϩʔυ൛ ▸ ౰೔Demoͱ͍ͯͨ͠ͱ͜Ζ͸ɺ͍͔ͭ͘εΫϦʔϯγϣοτΛషΓ෇͚ ·ͨ͠ɻ ▸ ͕࣌ؒ͋ͬͨΒ͠Ό΂Ζ͏ͱࢥͬͨεϥΠυ͕ޙΖʹ෇͚ͯ͋Γ·͢ɻ

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ࣗݾ঺հ ͋ Μ ͨ ͊ ୭ ͳ ͊

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๺㝳೗๏ ▸ ͖͍ͨͩΏ͖ͷΓ ▸ ʮ೗๏ʯΛԻಡΈ(༗৬ಡΈ)ͯ͠ňʹΐ΄͏ŉ ▸ ϋΠύɾϝσΟΞɾτϥϯεϨʔλ ▸ ೔༵ϓϩάϥϚ

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iPad ޿ౡห ݕࡧ ৄ͘͠͸

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ʮ͜Μͳײ͡ͷ͕ग़͖ͯ·͢ɻʯ

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(͜͜Ͱ࠷ۙͷଞͷ׆ಈΛը૾(ུ)෇͖Ͱ͝঺հͨ͠) ▸ ཆ໋ञ΢Σϒಈը޿ࠂ ެࣜwwwww ୈ26ճ೔ຊϓϩϞʔγϣϯاըίϯςετͰάϥ ϯϓϦ৆wwww ▸ όϑΝϦϯ ΢Σϒಈը޿ࠂ ެࣜwwwww ▸ ؟ڸࢢ৔ ެࣜww ςϨϏCM wwww ▸ ؟ڸࢢ৔ TVCM (2018) Χʔϓww Τϧwww ▸ ؟ڸࢢ৔ TVCM (2019) ϑϥϯεΞwwwwwww ▸ ࠓ೥ RCCதࠃ์ૹ(TBSܥ) ʰΠϚͳ·ͬ!ʱίϝϯςʔλwwww

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͋ͱ਺ֶ͕ ޷͖

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ͱ͍͏Θ͚ Ͱ

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SageMathͰ ਺ֶͷ ྗΛ ্͛·͢

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SageMathͰ ਺ֶͷ ྗΛ ্͛·͢

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SageMathͰ ਺ֶͷ ྗΛ ্͛·͢ ͞ ͛ · ͢ ͋ ͛ · ͢ ʊਓਓਓਓਓਓʊ ʼɹμδϟϨɹʻ ʉ̮̮̮̮̮̮ʉ

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SageMathͰ ਺ֶͷ ྗΛ ্͛·͢

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SageMathͰ ਺ֶͷ ྗΛ ্͛ΒΕͨΒɺ

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(ɾ㱼ɾ) ŘŘ!!

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͝ਗ਼ௌ͋Γ͕ͱ͏͍͟͝·ͨ͠ɻ

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΋͏ͪΐͬͱ ৄ͘͠

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ຊ೔ͷϝχϡʔ 4BHFͱ͸ 4BHFΛେֶͷतۀͰซ༻ ༡ͼ΍ݚڀʹ

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օ͞Μ͸ ਺ֶେ޷͖

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Sageͱ͸

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Sage ▸ Φʔϓϯιʔεͷ਺ֶιϑτ΢ΣΞγεςϜ ▸ ͍ΖΜͳ਺ֶιϑτ΢ΣΞΛPythonͰ౷߹ ▸ GAP, Maxima, NumPy, SciPy, mathplotlib, Sympy, FLINT, R ▸ ϛογϣϯ: Mathematica ͱ͔ Maple, Matlab, Magmaͷ୅ସΛ࡞Δ͜ͱ

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਺ࣜॲཧγε ςϜ

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਺ࣜॲཧγεςϜ͸਺ࣜΛ਺ࣜͷ··ѻ͑Δ ▸ ුಈখ਺఺਺ͷۙࣅͰ͸ͳ͘୅਺ܭࢉ͕ͦͷ··Ͱ͖Δ ▸ ԁप཰Λͦͷ·· ▸ ؔ਺ͷඍ෼ੵ෼ͷΑ͏ͳ΋ͷͱ͔ ▸ ͳͲ

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؆୯ʹࢼ͢

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؆୯ʹࢼ͢ ▸ ΢Σϒ ▸ Sage Cell Server https://sagecell.sagemath.org/ ▸ CoCalc (چSageMathCloud) https://cocalc.com ▸ Πϯετʔϧ ▸ ϩʔΧϧʹ ▸ ίϚϯυϥΠϯͰ΋

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CoCalc ͍͢͝ɻ ▸ SageNotebook ▸ Jupyter Notebook ▸ Rmarkdown ▸ ͚ͩ͡Όͳ͍ ▸ X11 ͱ͔΋Ͱ͖Δ

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Demo

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͜Μͳײ͡ͷ΋ͷΛ ͓ݟͤ͠·ͨ͠ɻ

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͜Μͳײ͡ͷ΋ͷΛ ͓ݟͤ͠·ͨ͠ɻ

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ຊ೔ͷϝχϡʔ 4BHFͱ͸ 4BHFΛେֶͷतۀͰซ༻ ༡ͼ΍ݚڀʹ

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ຊ೔ͷϝχϡʔ 4BHFͱ͸ 4BHFΛେֶͷतۀͰซ༻ ༡ͼ΍ݚڀʹ

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SageΛେֶͷ तۀͰซ༻

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SageΛେֶͷतۀͰซ༻ ▸ ޿ౡେֶͷڭҭֶ෦Ͱ਺ֶΛڭ͍͑ͯΔ ▸ ୅਺ֶܥͷೖ໳ߨٛͷҰ෦ Sage Λ࢖༻ͨ͠ ▸ ڭҭֶ෦Ͱɺதɾߴͷ਺ֶՊͷڭһ໔ڐΛऔΖ͏ͱ͢Δֶੜ ▸ ͕ͬͭΓϓϩάϥϛϯάΛ͢Δਓ޲͚Ͱ͸ͳ͍ ▸ ͪΐͬͱίϚϯυΛଧͬͯΈͯܭࢉͤ͞Δ͙Β͍Ͱे෼ศར ▸ ʮߴڃి୎ʯͱͯ͠

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ྫ

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ྫ ▸ ഒ਺ is Կ? ▸ ໿਺ is Կ? ▸ ࠷େެ໿਺ is Կ?

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࠷େެ໿਺ ▸ ࠷େެ໿਺ greatest common divisor (GCD) ▸ (1) 12, 18 ͷ࠷େެ໿਺͸? ▸ (2) 123456789 ͱ 987654321 ͷ࠷େެ໿਺͸? ▸ ΍Γํ ▸ ໿਺Λશ෦͍͋͛ͯ͘ ▸ ૉҼ਺෼ղͯ͠ڞ௨ͷૉҼ਺Λ͔͚Δ ▸ ϢʔΫϦουͷޓআ๏

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GCD΍Γํ͍Ζ͍Ζ ▸ ·ͣ͸खܭࢉͰख৮ΓײΛ ▸ खʹෛ͑ͳ͍ن໛͸ɺͦͷ͋ͱܭࢉػ࣮ݧ

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Demo

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͜Μͳײ͡ͷ΋ͷΛ ͓ݟͤ͠·ͨ͠ɻ ૉҼ਺෼ղͷॏ͞ͷײ֮

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͜͏͍͏ͱ͜Ζ͕ 4BHFΒ͍͠ ਺ֶιϑτ΢ΣΞΒ͍͠ XIJMFɻ OFYU@QSJNF ਺ Ͱ਺ΑΓେ͖͍ૉ਺Λฦ͢ ‏ʮΊͪΌͪ͘Ό଎͍ɻԿ͔͋Γͦ͏ͩͳʯͱٯʹࢥ͏

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ΞϧΰϦζϜͷҧ͍ ▸ खܭࢉʹՃ͑ͯ ▸ ܭࢉػ࣮ݧʹΑΓॏ͞ʹख৮ΓײΛಘΔ

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ָ͍͠!

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ຊ೔ͷϝχϡʔ 4BHFͱ͸ 4BHFΛେֶͷतۀͰซ༻ ༡ͼ΍ݚڀʹ

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ຊ೔ͷϝχϡʔ 4BHFͱ͸ 4BHFΛେֶͷतۀͰซ༻ ༡ͼ΍ݚڀʹ

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༡ͼ΍ݚڀ ʹ

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༡ͼ΍ݚڀʹSageΛ࢖͏࿩ ▸ ྫ Twitter ΑΓ

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༡ͼ΍ݚڀʹSageΛ࢖͏࿩ ▸ ྫ Twitter ΑΓ

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༡ͼ΍ݚڀʹSageΛ࢖͏࿩ ▸ ྫ Twitter ΑΓ ▸ ࣗવ਺ ʹରͯ͠    ͕੔਺ͷͱ͖ ͸ԿͰ͠ΐ͏͔ n 12502 + n3 n

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͜͏͍͏ͷΛݟΔͱʮզʑʯ͸ɺ ʮପԁۂઢͷ࿩ͩͳʯ ͱɺࢥ͏ɻ ʮ͋ɺପԁۂઢͷ࿩ͩͳʯ

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ପԁۂઢ?

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ପԁۂઢ ▸ (ॏࠜͳ͠) Ͱఆٛ͞ΕΔʮۂઢʯ ▸ ΋ͷ͘͢͝ݚڀ͞Ε͍ͯΔɻݱࡏ΋ɻ ▸ ਺ֶతʹ͘͢͝ັྗతͳର৅ ▸ ઌ΄Ͳͷ໰୊ͩͱɺ ▸ Λ ͱ͓͍ͨΒɺ ▸ ͱ͍͏ପԁۂઢͷ্ͷ֨ࢠ఺ (੔਺఺) (࠲ඪ͕੔਺ʹͳ͍ͬͯΔ ఺) (ͷ͏ͪ ͷ΋ͷ) ΛٻΊΔ໰୊ y2 = xͷ3࣍ࣜ 12502 + n3 y y2 = 12502 + n3 y2 = x3 + 12502 x > 0

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ପԁۂઢ ▸ ͷ஥ؒͨͪ y2 = x3 + 12502 -4 -2 0 2 4 -4 -2 0 2 4 y2 = x3 − x -4 -2 0 2 4 -4 -2 0 2 4 y2 = x3 + 1

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ପԁۂઢ ▸ ͷ஥ؒͨͪ ▸ ͜Ε͸ ͕࣮਺Λಈ͍ͨਤ ▸ ͜ͷதͰ ͕ͱ΋ʹ੔਺ʹͳΔ఺Λ୳͢ ▸ ࣮͸δʔήϧͷఆཧʮପԁۂઢͷ੔਺ղ͸ৗʹ༗ݶݸʯ y2 = x3 + 12502 x, y x, y -4 -2 0 2 4 -4 -2 0 2 4 y2 = x3 − x -4 -2 0 2 4 -4 -2 0 2 4 y2 = x3 + 1

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SageͰପԁۂઢΛѻ͏ ▸ Λ΍Δ ▸ Weierstrassํఔࣜ ͩͱ EllipticCurve([a1,a2,a3,a4,a6]) Ͱ࡞ΕΔɻ ▸ ͷ লུܗ: EllipticCurve([a4,a6]) ▸ E = EllipticCurve([0,0,0,0,1562500]) ▸ E.integral_points() y2 = x3 + 12502 y2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 a1 = a2 = a3 = 0

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Demo

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verbose=True ͢Δͱ్தͲΜͳܭࢉΛ͔ͨ͠ڭ͑ͯ͘Ε·͢

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Model-Weil܈(આ໌ུ)ͷੜ੒ݩ(આ໌ུ)ͱ͔΋Θ͔Δ

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SageͰପԁۂઢΛѻ͏ ▸ E.integral_points() ▸ [(-100 : 750 : 1),  (0 : 1250 : 1),  (125 : 1875 : 1),  (500 : 11250 : 1),  (600 : 14750 : 1),  (66500 : 17148750 : 1)] ▸ ౴͑ n = 125, 500, 600, 66500

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ָ͍͠ ✌('ω'✌ )ࡾ✌('ω')✌ࡾ( ✌'ω')✌

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·ͱΊ ▸ SageΛ࢖͏ͱ ▸ ਺ֶͷɺڭҭɾֶशɾݚڀͰ ▸ ໾ཱͭɾ༡΂Δ ͋Γ͕ͱ͏͍͟͝·ͨ͠ɻ

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͔͜͜Β͸͕࣌ؒ͋Ε͹͓·͚ ▸ scheme ͱ͔ generic point ͱ͔Ψνͷ୅਺زԿͷΫϥε΋༻ҙ͞Ε͍ͯ Δɻ ▸ Try: ݚڀͰ࢖͓͏ͱͯ͠։ൃʹߩݙ͍ͨ͠ͱ͜Ζ͕Ͱ͖ͯͨͷͰߩݙ͢Δ ▸ ྫ΋͏Ұͭ

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༡ͼ΍ݚڀʹSageΛ࢖͏࿩ ▸ ྫ Twitter ΑΓ (·ͨTwitter͔wwww)

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Demo

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Python 3 ରԠͲ͏ͳͷ? ▸ ରԠ͍ͯ͠Δɻ ▸ Python3-compatible code - Sage Wiki https://wiki.sagemath.org/Python3-compatible%20code ࢀর ▸ ࠓޙ͸όΠφϦ΋ఏڙ༧ఆɻversion 8.8 ͷޙ͔ΒΒ͍͠ͷͰͦΖͦΖɻ ▸ CoCalcͰ͸ΧʔωϧͰબ΂·͢ɻ ▸ खಈϏϧυ͸  $ make configure  $ ./configure --with-python=3  $ make build  ͰOK

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ݱ୅਺ֶ΋͔ͳΓॆ࣮ ▸ scheme ͱ͔ generic point ͱ͔Ψνͷ୅਺زԿͷΫϥε΋༻ҙ͞Ε͍ͯΔ ▸ Points on projective varieties, Scheme morphism for points on projective varieties https:// github.com/sagemath/sage/blob/master/src/sage/schemes/projective/projective_point.py ▸ P = ProjectiveSpace(2, QQ) ←༗ཧ਺܎਺ͷࣹӨฏ໘͕͜ΕͰͰ͖Δ ▸ P(2, 3/5, 4) ←Pͷ্ͷ੪࣍࠲ඪ (2 : 3/5 : 4) ͷ఺ͷΦϒδΣΫτ͕͜ΕͰͰ͖Δ ▸ ࣹӨۭؒͷ఺ΛྻڍରԠʹ͔ͨͬͨ͠ɻ ▸ → ࠓޙ͸։ൃʹࢀՃ͍ͨ͠ɻ ▸ ChangeLog ΛݟΔͱ৽͍͠ίϯτϦϏϡʔλ͕ଟ਺!