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Θ͔ͬͨͭ΋ΓʹͳΔ Gröbnerجఈ ੴҪ େւ 13೥5݄4೔౔༵೔

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ࣗݾ঺հ ੴҪେւʢa.k.a. @mr_konnʣ Twitter: @mr_konn Blog: http:/ /blog.konn-san.com/ Web: http:/ /konn-san.com/ Haskell lover ૣҴాେֶ਺ֶՊ࢛೥ ຊ౰͸ϩδοΫͷਓؒͷഺ͕ࠓ೥͸ԑ͋ͬͯ୅ ਺زԿʢಛʹ Gröbner جఈʣ ͷݚڀࣨʹ computational-algebra ͷ࡞ऀ 13೥5݄4೔౔༵೔

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Table of Contents Gröbner جఈͷ؆୯ͳಋೖ Gröbner جఈͷܭࢉ๏ Buchberger ΞϧΰϦζϜ ࠷దԽख๏ɿsyzygy, sugar strategy Haskell ͷܕϨϕϧϓϩάϥϛϯάͷݱࡏ 13೥5݄4೔౔༵೔

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Table of Contents Gröbner جఈͷ؆୯ͳಋೖ Gröbner جఈͷܭࢉ๏ Buchberger ΞϧΰϦζϜ ࠷దԽख๏ɿsyzygy, sugar strategy Haskell ͷܕϨϕϧϓϩάϥϛϯάͷݱࡏ 13೥5݄4೔౔༵೔

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Gröbner جఈ ͷ؆୯ͳಋೖ 13೥5݄4೔౔༵೔

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Gröbner جఈͷԠ༻ ߴ࣍࿈ཱํఔࣜͷจࣈফڈ ॳ౳زԿͷࣗಈূ໌ ౷ܭʹ΋Կ͔Ԡ༻͕͋ΔΒ͍͠ ϩϘςΟΫε ܭࢉػ୅਺ɺ୅਺زԿֶ 13೥5݄4೔౔༵೔

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Gröbner جఈ 13೥5݄4೔౔༵೔

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Gröbner جఈ “άϨϒφʔ͖͍ͯ” ͱಡΉ 13೥5݄4೔౔༵೔

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Gröbner جఈ “άϨϒφʔ͖͍ͯ” ͱಡΉ 1960s: ኍதฏ༞ɺB. Buchberger Β͕ಠཱʹൃݟ 13೥5݄4೔౔༵೔

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Gröbner جఈ “άϨϒφʔ͖͍ͯ” ͱಡΉ 1960s: ኍதฏ༞ɺB. Buchberger Β͕ಠཱʹൃݟ Gröbner ͸ Buchberger ͷࢣঊͷ໊લ 13೥5݄4೔౔༵೔

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Gröbner جఈ “άϨϒφʔ͖͍ͯ” ͱಡΉ 1960s: ኍதฏ༞ɺB. Buchberger Β͕ಠཱʹൃݟ Gröbner ͸ Buchberger ͷࢣঊͷ໊લ ኍத͸ಛҟ఺ղফͷͨΊʹߟ͑ͨ 13೥5݄4೔౔༵೔

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Gröbner جఈ “άϨϒφʔ͖͍ͯ” ͱಡΉ 1960s: ኍதฏ༞ɺB. Buchberger Β͕ಠཱʹൃݟ Gröbner ͸ Buchberger ͷࢣঊͷ໊લ ኍத͸ಛҟ఺ղফͷͨΊʹߟ͑ͨ ٕज़ͷਐาͰܭࢉग़དྷΔΑ͏ʹͳΓԠ༻͕ग़Δ 13೥5݄4೔౔༵೔

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Gröbner جఈ “άϨϒφʔ͖͍ͯ” ͱಡΉ 1960s: ኍதฏ༞ɺB. Buchberger Β͕ಠཱʹൃݟ Gröbner ͸ Buchberger ͷࢣঊͷ໊લ ኍத͸ಛҟ఺ղফͷͨΊʹߟ͑ͨ ٕज़ͷਐาͰܭࢉग़དྷΔΑ͏ʹͳΓԠ༻͕ग़Δ ମ্ଟ߲ࣜ؀ͷΠσΞϧͷੑ࣭ͷΑ͍ੜ੒ݩ 13೥5݄4೔౔༵೔

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ମ্ଟ߲ࣜ؀ͷΠσΞϧ ͷੑ࣭ͷΑ͍ੜ੒ݩ 13೥5݄4೔౔༵೔

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ʁʁʁ 13೥5݄4೔౔༵೔

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ମ্ଟ߲ࣜ؀ͷΠσΞϧ ͷੑ࣭ͷΑ͍ੜ੒ݩ 13೥5݄4೔౔༵೔

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ମ্ଟ߲ࣜ؀ͷΠσΞϧ ͷੑ࣭ͷΑ͍ੜ੒ݩ 13೥5݄4೔౔༵೔

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ମ ͍ͨʢ≠͔ΒͩʣͱಡΉ ଍͠ࢉɺҾ͖ࢉɺֻ͚ࢉɺׂΓࢉʢ0Ҏ֎ʣ͕ग़͖ΔΑ ͏ͳߏ଄ ྫɿ༗ཧ਺ମ ℚ ͱ͔ෳૉ਺ମ ℂ ͱ͔ ྫɿ੔਺શମ ℤ ͸ମͰͳ͍ʢ1/2͸੔਺͡Όͳ͍ʣ ҎԼͰ͸খจࣈͷ k ͰମΛද͢ 13೥5݄4೔౔༵೔

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ମ্ଟ߲ࣜ؀ͷΠσΞϧ ͷੑ࣭ͷΑ͍ੜ੒ݩ 13೥5݄4೔౔༵೔

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ମ্ͷଟ߲ࣜ؀ k[X1 , …, Xn ] = {k Λ܎਺ͱ͢Δ n-ม਺ଟ߲ࣜશମ} ม਺ͷ਺͕ࣗ໌ͳ࣌͸ k[] = k[X1 , …, Xn ] ͱॻ͘ ࣗવʹ଍͠ࢉɺҾ͖ࢉɺֻ͚ࢉ͕ఆٛͰ͖Δʢׂ Γࢉ͸ग़དྷΔͱ͸ݶΒͳ͍ʣ ྫɿX1 2 + 2 X1 X2  + X2 2 ∈ k[X1 , X2 ], y2 − x ∈ k[x, y] 13೥5݄4೔౔༵೔

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ମ্ଟ߲ࣜ؀ͷΠσΞϧ ͷੑ࣭ͷΑ͍ੜ੒ݩ 13೥5݄4೔౔༵೔

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ΠσΞϧ ؀ R ͷ෦෼ू߹ I ͕ ΠσΞϧ Ͱ͋Δͱ͸ 1. ೚ҙͷ x, y ∈ I ʹ͍ͭͯ x − y ∈ I 2.೚ҙͷ c ∈ R, x ∈ I ʹ͍ͭͯ cx ∈ I ഒ਺֓೦ͷ҃ΔछͷҰൠԽʹͳ͍ͬͯΔ ΠσΞϧ I ʹଐ͢Δ ⇔ I ͷݩͰׂΕΔ a ͕ b ͷഒ਺ ⇔ ⟨a⟩ ⊆ ⟨b⟩ ఆٛ 13೥5݄4೔౔༵೔

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ͳͥΠσΞϧʁ 13೥5݄4೔౔༵೔

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ͳͥΠσΞϧʁ ΠσΞϧ͸زԿతʹ͸ͦͷྵ఺ͷ੒͢ਤܗͱରԠ͢Δ ʢਖ਼֬ʹ͸ࠜجΠσΞϧ͕ରԠʣ 13೥5݄4೔౔༵೔

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ͳͥΠσΞϧʁ ΠσΞϧ͸زԿతʹ͸ͦͷྵ఺ͷ੒͢ਤܗͱରԠ͢Δ ʢਖ਼֬ʹ͸ࠜجΠσΞϧ͕ରԠʣ ⟨y − x2⟩ɿ์෺ઢ 13೥5݄4೔౔༵೔

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ͳͥΠσΞϧʁ ΠσΞϧ͸زԿతʹ͸ͦͷྵ఺ͷ੒͢ਤܗͱରԠ͢Δ ʢਖ਼֬ʹ͸ࠜجΠσΞϧ͕ରԠʣ ⟨y − x2⟩ɿ์෺ઢ ⟨y2 + x2 − 1, y + 2x − 1⟩ɿԁͱ௚ઢͷަ఺ 13೥5݄4೔౔༵೔

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ͳͥΠσΞϧʁ ΠσΞϧ͸زԿతʹ͸ͦͷྵ఺ͷ੒͢ਤܗͱରԠ͢Δ ʢਖ਼֬ʹ͸ࠜجΠσΞϧ͕ରԠʣ ⟨y − x2⟩ɿ์෺ઢ ⟨y2 + x2 − 1, y + 2x − 1⟩ɿԁͱ௚ઢͷަ఺ ަ఺ 㱻 ࿈ཱํఔࣜͷղ 13೥5݄4೔౔༵೔

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ͳͥΠσΞϧʁ ΠσΞϧ͸زԿతʹ͸ͦͷྵ఺ͷ੒͢ਤܗͱରԠ͢Δ ʢਖ਼֬ʹ͸ࠜجΠσΞϧ͕ରԠʣ ⟨y − x2⟩ɿ์෺ઢ ⟨y2 + x2 − 1, y + 2x − 1⟩ɿԁͱ௚ઢͷަ఺ ަ఺ 㱻 ࿈ཱํఔࣜͷղ ΠσΞϧ͸࿈ཱ͞Εͨؔ܎ࣜͱ΋ݟΔ͜ͱ͕ग़དྷΔ 13೥5݄4೔౔༵೔

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ͳͥΠσΞϧʁ ΠσΞϧ͸زԿతʹ͸ͦͷྵ఺ͷ੒͢ਤܗͱରԠ͢Δ ʢਖ਼֬ʹ͸ࠜجΠσΞϧ͕ରԠʣ ⟨y − x2⟩ɿ์෺ઢ ⟨y2 + x2 − 1, y + 2x − 1⟩ɿԁͱ௚ઢͷަ఺ ަ఺ 㱻 ࿈ཱํఔࣜͷղ ΠσΞϧ͸࿈ཱ͞Εͨؔ܎ࣜͱ΋ݟΔ͜ͱ͕ग़དྷΔ Gröbner جఈΛ࢖͏ͱ৭Μͳਤܗɾ࿈ཱํఔࣜʹ ؔ͢Δܭࢉ͕؆୯ʹग़དྷΔΑ͏ʹͳΔ 13೥5݄4೔౔༵೔

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ମ্ଟ߲ࣜ؀ͷΠσΞϧ ͷੑ࣭ͷΑ͍ੜ੒ݩ 13೥5݄4೔౔༵೔

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ੜ੒ݩ ্ͷ ⟨f1 ,…, fr ⟩ ͸͖ͬ͞ͷΠσΞϧͷఆٛΛຬͨ͢ɻ ٯʹɺମ্ଟ߲ࣜ؀ͷΠσΞϧ͸ɺద౰ͳ f1 ,…, fr ʹ ΑΓɺ͢΂ͯ͜ͷܗͰॻ͚ΔʢHilbert ͷجఈఆཧʣ f1 , … , fr ∈ k[ ] ʹ͍ͭͯɺ ⟨f1 ,…, fr ⟩ := {a1 f1  +…+ ar fr ∣ ai  ∈ k[] (i = 1, … r)} Λ f1 ,…, fr ʹΑΓੜ੒͞ΕΔΠσΞϧͱ͍͏ɻ I = ⟨f1 ,…, fr ⟩ ͷ࣌ɺ f1 ,…, fr Λ I ͷجఈɺ·ͨ͸ ੜ੒ݩͱݺͿɻ ఆٛ 13೥5݄4೔౔༵೔

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Hilbert ͷجఈఆཧ Noether ؀্ͷଟ߲ࣜ؀͸ Noether ؀Ͱ͋Δ ఆཧ 13೥5݄4೔౔༵೔

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Hilbert ͷجఈఆཧ Noether ؀ͱ͸ʮ೚ҙͷΠσΞϧ͕༗ݶݸͷੜ੒ݩͰॻ͚Δʢʹ༗ ݶੜ੒ʣʯͱ͍͏৚݅Λຬͨ͢؀ Noether ؀্ͷଟ߲ࣜ؀͸ Noether ؀Ͱ͋Δ ఆཧ 13೥5݄4೔౔༵೔

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Hilbert ͷجఈఆཧ Noether ؀ͱ͸ʮ೚ҙͷΠσΞϧ͕༗ݶݸͷੜ੒ݩͰॻ͚Δʢʹ༗ ݶੜ੒ʣʯͱ͍͏৚݅Λຬͨ͢؀ ܭࢉػͰѻ͏ʹ͸ɺଟ߲ࣜ؀ͷΠσΞϧ͸༗ݶݸͷଟ߲ࣜͷ૊ͱ͠ ͯఆٛ͢Ε͹े෼ Noether ؀্ͷଟ߲ࣜ؀͸ Noether ؀Ͱ͋Δ ఆཧ 13೥5݄4೔౔༵೔

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Hilbert ͷجఈఆཧ Noether ؀ͱ͸ʮ೚ҙͷΠσΞϧ͕༗ݶݸͷੜ੒ݩͰॻ͚Δʢʹ༗ ݶੜ੒ʣʯͱ͍͏৚݅Λຬͨ͢؀ ܭࢉػͰѻ͏ʹ͸ɺଟ߲ࣜ؀ͷΠσΞϧ͸༗ݶݸͷଟ߲ࣜͷ૊ͱ͠ ͯఆٛ͢Ε͹े෼ ূ໌͸Ή͔͍͕ͣ͠ɺମͷ৔߹ʹ͍ͭͯ͸ Gröbner جఈΛ࢖͏ͱ ؆୯ʹࣔͤΔɻ ʢূ໌ɿGröbner جఈΛऔΕ͹Α͍■ʣ Noether ؀্ͷଟ߲ࣜ؀͸ Noether ؀Ͱ͋Δ ఆཧ 13೥5݄4೔౔༵೔

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ମ্ଟ߲ࣜ؀ͷΠσΞϧ ͷੑ࣭ͷΑ͍ੜ੒ݩ 13೥5݄4೔౔༵೔

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Α͍ੑ࣭ʁ ࠨͷ໰୊Λߟ͑Δ f1 ͱ f2 ͷ “࠷େެ໿ࣜ” f = GCD(f1 ,  f2 ) ͕Θ ͔Ε͹Α͍ I = ⟨ f ⟩ ͱͳΔͷͰɺ ͋ͱ͸ g, h Λ f Ͱׂ Γ੾ΕΔ͔ݟΔ f1  = x3 + x2 + x + 1 f2  = x2 − x − 2 I = ⟨ f1 ,  f2 ⟩ ⊆ k[x] ͱ͢Δͱɺg = x3 − 1,  h = x2 + 1 ͸ͦΕͧΕ I ʹଐ͔͢ʁ ໰୊ 1 13೥5݄4೔౔༵೔

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࠷େެ໿ࣜͷٻΊํ Euclid ͷޓআ๏Λ࢖͏ ੔਺͚ͩͰͳ͘ɺׂΓࢉݪཧ͕੒ཱ͢Δ೚ ҙͷ؀Ͱ࢖͑Δ ੔਺ͷ৔߹ɿ͋Δݩ͕ ⟨a, b⟩ ʹଐ͢Δ ɹ㱻 a, b ͰׂΕΔ ɹ㱻 a, b ͷ࠷େެ໿਺ͰׂΕΔ ಉ༷ͷ͜ͱ͕Ұม਺ଟ߲ࣜ؀Ͱ΋੒ཱ 13೥5݄4೔౔༵೔

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Euclid ͷޓআ๏ ೖྗɿf, g 1. f  Λ g ͰׂΓࢉͨ͠༨ΓΛ r0 ͱ͢Δ 2.g  Λ r0 ͰׂΓࢉͨ͠༨ΓΛ r1 ͱ͢Δ 3.r0   Λ r1 ͰׂΓࢉͨ͠༨ΓΛ r2 ͱ͢Δ 4.ҎԼθϩʹͳΔ·Ͱ܁Γฦ͠ ग़ྗɿθϩͰͳ͍࠷ޙͷ rn 13೥5݄4೔౔༵೔

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Ұม਺ଟ߲ࣜͷׂΓࢉ ઌఔͷ໰୊Ͱ Euclidͷ ޓআ๏Λ࢖͏ͱ͖ͷ࠷ ॳͷεςοϓ ߱ႈͷॱʹฒ΂Δ େ͖͍߲ͱͷா৲Λ߹ Θ͍ͤͯ͘ x +2 x2 x 2 x3 +x2 +x +1 x3 x2 2x 2x2 +3x +1 2x2 2x 4 5x +5 13೥5݄4೔౔༵೔

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͜ΕΛଟม਺ʹ ҰൠԽͰ͖ͳ͍͔ʁ 13೥5݄4೔౔༵೔

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ଟม਺ͷ৔߹ 13೥5݄4೔౔༵೔

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ଟม਺ͷ৔߹ f1  = X2Y − 1 f2  = X3 − Y2 − X I = ⟨ f1 ,  f2 ⟩ ⊆ k[X, Y] ͱ͢Δͱɺ g = X2Y2 − X3Y − XY − 1 ͸ I ʹଐ͔͢ʁ ໰୊ 2 13೥5݄4೔౔༵೔

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ଟม਺ͷ৔߹ ೋม਺ͷ৔߹Λߟ͑Δ f1  = X2Y − 1 f2  = X3 − Y2 − X I = ⟨ f1 ,  f2 ⟩ ⊆ k[X, Y] ͱ͢Δͱɺ g = X2Y2 − X3Y − XY − 1 ͸ I ʹଐ͔͢ʁ ໰୊ 2 13೥5݄4೔౔༵೔

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ଟม਺ͷ৔߹ ೋม਺ͷ৔߹Λߟ͑Δ f1 ͱ f2 ͷ “࠷େެ໿ࣜ” ͷΑ͏ͳ෺Λߟ͑ΒΕ Δ͔ʁ f1  = X2Y − 1 f2  = X3 − Y2 − X I = ⟨ f1 ,  f2 ⟩ ⊆ k[X, Y] ͱ͢Δͱɺ g = X2Y2 − X3Y − XY − 1 ͸ I ʹଐ͔͢ʁ ໰୊ 2 13೥5݄4೔౔༵೔

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ଟม਺ͷ৔߹ ೋม਺ͷ৔߹Λߟ͑Δ f1 ͱ f2 ͷ “࠷େެ໿ࣜ” ͷΑ͏ͳ෺Λߟ͑ΒΕ Δ͔ʁ f1  = X2Y − 1 f2  = X3 − Y2 − X I = ⟨ f1 ,  f2 ⟩ ⊆ k[X, Y] ͱ͢Δͱɺ g = X2Y2 − X3Y − XY − 1 ͸ I ʹଐ͔͢ʁ ໰୊ 2 ➡ ग़དྷͳ͍ʂ 13೥5݄4೔౔༵೔

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࠷େެ໿ࣜ͸औΕͳ͍ 13೥5݄4೔౔༵೔

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࠷େެ໿ࣜ͸औΕͳ͍ ࠷େެ໿͕ࣜඞͣऔΕΔͳΒɺ೚ҙͷΠσΞ ϧ͸Ұͭͷଟ߲ࣜʹΑΓ⟨ f ⟩ ͷܗͰॻ͚Δ 13೥5݄4೔౔༵೔

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࠷େެ໿ࣜ͸औΕͳ͍ ࠷େެ໿͕ࣜඞͣऔΕΔͳΒɺ೚ҙͷΠσΞ ϧ͸Ұͭͷଟ߲ࣜʹΑΓ⟨ f ⟩ ͷܗͰॻ͚Δ ͔͠͠ɺྫ͑͹⟨x, y⟩ʹ্͍ͭͯͷΑ͏ͳ f ͸ଘࡏ͠ͳ͍ʂ 13೥5݄4೔౔༵೔

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࠷େެ໿ࣜ͸औΕͳ͍ ࠷େެ໿͕ࣜඞͣऔΕΔͳΒɺ೚ҙͷΠσΞ ϧ͸Ұͭͷଟ߲ࣜʹΑΓ⟨ f ⟩ ͷܗͰॻ͚Δ ͔͠͠ɺྫ͑͹⟨x, y⟩ʹ্͍ͭͯͷΑ͏ͳ f ͸ଘࡏ͠ͳ͍ʂ ผͷखஈΛߟ͑Δ΂͠ 13೥5݄4೔౔༵೔

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࠷େެ໿ࣜ͸औΕͳ͍ ࠷େެ໿͕ࣜඞͣऔΕΔͳΒɺ೚ҙͷΠσΞ ϧ͸Ұͭͷଟ߲ࣜʹΑΓ⟨ f ⟩ ͷܗͰॻ͚Δ ͔͠͠ɺྫ͑͹⟨x, y⟩ʹ্͍ͭͯͷΑ͏ͳ f ͸ଘࡏ͠ͳ͍ʂ ผͷखஈΛߟ͑Δ΂͠ ׂΓࢉͷ༨Γ͕ॏཁͩͬͨ 13೥5݄4೔౔༵೔

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࠷େެ໿ࣜ͸औΕͳ͍ ࠷େެ໿͕ࣜඞͣऔΕΔͳΒɺ೚ҙͷΠσΞ ϧ͸Ұͭͷଟ߲ࣜʹΑΓ⟨ f ⟩ ͷܗͰॻ͚Δ ͔͠͠ɺྫ͑͹⟨x, y⟩ʹ্͍ͭͯͷΑ͏ͳ f ͸ଘࡏ͠ͳ͍ʂ ผͷखஈΛߟ͑Δ΂͠ ׂΓࢉͷ༨Γ͕ॏཁͩͬͨ ➡ ׂΓࢉΛҰൠԽͯ͠ΈΑ͏ʂ 13೥5݄4೔౔༵೔

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ׂΓࢉͷҰൠԽ 13೥5݄4೔౔༵೔

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ׂΓࢉͷҰൠԽ ෳ਺ͷଟ߲ࣜͰҰͭͷଟ߲ࣜΛׂΖ͏ʂ ଟ߲ࣜΛฒ΂͓͍ͯͯॱʹׂΕͳ͍͔ࢼ͢ Ұม਺ଟ߲ࣜͷ৔߹ͷ߱ႈͷॱͷΑ͏ʹɺԿ ͔౎߹ͷΑ͍ฒ΂ํΛߟ͑Δඞཁ͕͋Δ 13೥5݄4೔౔༵೔

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ׂΓࢉͷҰൠԽ ෳ਺ͷଟ߲ࣜͰҰͭͷଟ߲ࣜΛׂΖ͏ʂ ଟ߲ࣜΛฒ΂͓͍ͯͯॱʹׂΕͳ͍͔ࢼ͢ Ұม਺ଟ߲ࣜͷ৔߹ͷ߱ႈͷॱͷΑ͏ʹɺԿ ͔౎߹ͷΑ͍ฒ΂ํΛߟ͑Δඞཁ͕͋Δ ➡ ୯߲ࣜॱংʂ 13೥5݄4೔౔༵೔

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ଟ߲ࣜͷදݱ 13೥5݄4೔౔༵೔

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ଟ߲ࣜͷදݱ ୯߲ࣜॱংͷલʹɺଟ߲ࣜͷදݱʹ͍ͭͯ 13೥5݄4೔౔༵೔

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ଟ߲ࣜͷදݱ ୯߲ࣜॱংͷલʹɺଟ߲ࣜͷදݱʹ͍ͭͯ จࣈͱࢦ਺ͷྻʢ1, x2, xy, y2 ͳͲʣΛ୯߲ࣜͱ͍ ͍ɺͦΕͧΕʹ܎਺Λॻ͚ͯՃ͑ͨ΋ͷΛଟ߲ࣜ ͱݺͿͷͩͬͨ 13೥5݄4೔౔༵೔

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ଟ߲ࣜͷදݱ ୯߲ࣜॱংͷલʹɺଟ߲ࣜͷදݱʹ͍ͭͯ จࣈͱࢦ਺ͷྻʢ1, x2, xy, y2 ͳͲʣΛ୯߲ࣜͱ͍ ͍ɺͦΕͧΕʹ܎਺Λॻ͚ͯՃ͑ͨ΋ͷΛଟ߲ࣜ ͱݺͿͷͩͬͨ จࣈͷؒʹॱ൪ΛܾΊΕ͹ɺ୯߲ࣜ͸੔਺ͷϕΫ τϧͱಉҰࢹग़དྷΔ 13೥5݄4೔౔༵೔

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ଟ߲ࣜͷදݱ ୯߲ࣜॱংͷલʹɺଟ߲ࣜͷදݱʹ͍ͭͯ จࣈͱࢦ਺ͷྻʢ1, x2, xy, y2 ͳͲʣΛ୯߲ࣜͱ͍ ͍ɺͦΕͧΕʹ܎਺Λॻ͚ͯՃ͑ͨ΋ͷΛଟ߲ࣜ ͱݺͿͷͩͬͨ จࣈͷؒʹॱ൪ΛܾΊΕ͹ɺ୯߲ࣜ͸੔਺ͷϕΫ τϧͱಉҰࢹग़དྷΔ x > y ͱ͢Ε͹ɺx2 ↔ (2, 0), xy ↔ (1, 1),  1 ↔ (0, 0) ͱݴ͏۩߹ʹରԠ͢Δ 13೥5݄4೔౔༵೔

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ଟ߲ࣜͷදݱ ୯߲ࣜॱংͷલʹɺଟ߲ࣜͷදݱʹ͍ͭͯ จࣈͱࢦ਺ͷྻʢ1, x2, xy, y2 ͳͲʣΛ୯߲ࣜͱ͍ ͍ɺͦΕͧΕʹ܎਺Λॻ͚ͯՃ͑ͨ΋ͷΛଟ߲ࣜ ͱݺͿͷͩͬͨ จࣈͷؒʹॱ൪ΛܾΊΕ͹ɺ୯߲ࣜ͸੔਺ͷϕΫ τϧͱಉҰࢹग़དྷΔ x > y ͱ͢Ε͹ɺx2 ↔ (2, 0), xy ↔ (1, 1),  1 ↔ (0, 0) ͱݴ͏۩߹ʹରԠ͢Δ ͦ͜Ͱ n-ม਺୯߲ࣜશମΛℤn ≥0 ͱಉҰࢹ͢Δ 13೥5݄4೔౔༵೔

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୯߲ࣜॱং ఆٛ 13೥5݄4೔౔༵೔

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୯߲ࣜॱং ℤn ≥0 ্ͷॱংؔ܎ > ͸ɺ࣍ͷࡾͭͷ৚݅Λຬͨ ͢ͱ͖୯߲ࣜॱংͰ͋Δͱݴ͏ɿ ఆٛ 13೥5݄4೔౔༵೔

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୯߲ࣜॱং ℤn ≥0 ্ͷॱংؔ܎ > ͸ɺ࣍ͷࡾͭͷ৚݅Λຬͨ ͢ͱ͖୯߲ࣜॱংͰ͋Δͱݴ͏ɿ 1. શॱংੑɿ (α < β) or (α = β) or (α > β) ఆٛ 13೥5݄4೔౔༵೔

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୯߲ࣜॱং ℤn ≥0 ্ͷॱংؔ܎ > ͸ɺ࣍ͷࡾͭͷ৚݅Λຬͨ ͢ͱ͖୯߲ࣜॱংͰ͋Δͱݴ͏ɿ 1. શॱংੑɿ (α < β) or (α = β) or (α > β) 2.Ճ๏ੑɿα ≤ β ⇒ α + γ ≤ β + γ ఆٛ 13೥5݄4೔౔༵೔

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୯߲ࣜॱং ℤn ≥0 ্ͷॱংؔ܎ > ͸ɺ࣍ͷࡾͭͷ৚݅Λຬͨ ͢ͱ͖୯߲ࣜॱংͰ͋Δͱݴ͏ɿ 1. શॱংੑɿ (α < β) or (α = β) or (α > β) 2.Ճ๏ੑɿα ≤ β ⇒ α + γ ≤ β + γ 3.ඇෛੑɿα ≥ 0 (೚ҙͷ α ∈ ℤn ≥0 ʹ͍ͭͯ) ఆٛ 13೥5݄4೔౔༵೔

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୯߲ࣜॱংͷิ଍ 13೥5݄4೔౔༵೔

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୯߲ࣜॱংͷิ଍ 3 ͷඇෛੑͷ৚݅͸ɺશॱংੑͱՃ๏ੑͷԼͰ ҎԼͷೋͭͱಉ஋ 13೥5݄4೔౔༵೔

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୯߲ࣜॱংͷิ଍ 3 ͷඇෛੑͷ৚݅͸ɺશॱংੑͱՃ๏ੑͷԼͰ ҎԼͷೋͭͱಉ஋ ੔ྻੑɿ೚ҙͷۭͰͳ͍ ℤn ≥0 ͷ෦෼ू߹͕ < ʹؔͯ͠࠷খ஋Λ࣋ͭ 13೥5݄4೔౔༵೔

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୯߲ࣜॱংͷิ଍ 3 ͷඇෛੑͷ৚݅͸ɺશॱংੑͱՃ๏ੑͷԼͰ ҎԼͷೋͭͱಉ஋ ੔ྻੑɿ೚ҙͷۭͰͳ͍ ℤn ≥0 ͷ෦෼ू߹͕ < ʹؔͯ͠࠷খ஋Λ࣋ͭ ߱࠯৚݅ɿ > ʹؔ͢ΔແݶԼ߱ྻ " " " α0  > α1  > α2  > ⋯ > αn  > ⋯ ͸ଘࡏ͠ͳ͍ 13೥5݄4೔౔༵೔

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୯߲ࣜॱংͷิ଍ 3 ͷඇෛੑͷ৚݅͸ɺશॱংੑͱՃ๏ੑͷԼͰ ҎԼͷೋͭͱಉ஋ ੔ྻੑɿ೚ҙͷۭͰͳ͍ ℤn ≥0 ͷ෦෼ू߹͕ < ʹؔͯ͠࠷খ஋Λ࣋ͭ ߱࠯৚݅ɿ > ʹؔ͢ΔແݶԼ߱ྻ " " " α0  > α1  > α2  > ⋯ > αn  > ⋯ ͸ଘࡏ͠ͳ͍ ͜ΕΒ͸ɺΞϧΰϦζϜͷఀࢭੑͷূ໌Ͱ࢖͏ 13೥5݄4೔౔༵೔

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୯߲ࣜॱংͷྫ ҎԼ α = (a1 , …, an ), β = (b1 , …, bn ) ͱ͢Δ ࣙॻࣜॱং >lex ͸୯߲ࣜॱং α >lex β :⟺ a1  = b1 , …, ai−1  = bi−1 ,  ai  > bi ٯࣙॻࣜॱং >revlex ͸୯߲ࣜॱংͰͳ͍ α >revlex β :⟺ ai+1  = bi+1 , …, an  = bn ,  ai  < bi ʢ※࠷ޙͷෆ౳߸ͷ޲͖ʹ஫ҙʂʣ 13೥5݄4೔౔༵೔

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୯߲ࣜॱংͷྫ ࣍਺෇͖ࣙॻࣜॱং >grlex ͸୯߲ࣜॱং α >grlex β :⟺ ∣α∣ > ∣β∣ or (∣α∣=∣β∣ and α >lex β) ͨͩ͠ ∣α∣ = a1  + ⋯ + an ʢશ࣍਺ʣ ࣍਺෇͖ٯࣙॻࣜॱং >grevlex ͸୯߲ࣜॱং α >grevlex β :⟺ ∣α∣>∣β∣ or (∣α∣=∣β∣ and α >revlex β) 13೥5݄4೔౔༵೔

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໿ଋ ୯߲ࣜॱং > Λݻఆ͢Δɻଟ߲ࣜ f ʹݱΕΔ ʢ܎਺ൈ͖ͷʣ୯߲ࣜͷதͰɺ> ʹ͍ͭͯ࠷ େͱͳΔ΋ͷΛઌ಄୯߲ࣜͱӠ͍ɺLM(f) Ͱ ද͢ɻLM(f) ͷ܎਺Λઌ಄܎਺ͱ͍͍ɺLC(f) Ͱද͢ɻLT(f) = LC(f) LM(f) Λઌ಄߲ͱݺͿɻ 13೥5݄4೔౔༵೔

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ྫ f = x + 2y2w + 3z3 (x > y > z >w) >lex ɿLM(f) = x, LC(f) = 1, LT(f) = x >grlex ɿLM(f) = y2w, LC(f) = 2, LT(f) = 2y2w >grevlex ɿLM(f) = z3, LC(f) = 3, LT(f) = 3z3 13೥5݄4೔౔༵೔

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ଟม਺ͷׂΓࢉ Ҏ্Λ༻͍ͯଟ߲ࣜ൛ͷׂΓࢉΛఆٛ͢Δ ୯߲ࣜॱংΛҰܾͭΊɺׂΔࣜɺׂΒΕΔࣜ ͱ΋ʹେ͖͍୯߲ࣜॱʹ੔ྻ͓ͯ͘͠ Ͳͷॱ൪ͰׂΔ͔͸ݻఆ͢Δʂ 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  a2  =  XY − 1 Y2 − 1 X2Y + XY2 + Y2 XY − 1, Y2 − 1 Ͱ X2Y + XY2 + Y2 ΛׂΔɻࣙॻࣜॱংΛ࢖͏ɻ 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  a2  =  XY − 1 Y2 − 1 X2Y + XY2 + Y2 ઌ಄߲ʹ஫໨ɻ࠷ॳͷࣜͰׂΕΔ 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  X a2  =  XY − 1 Y2 − 1 X2Y + XY2 + Y2 X2Y − X X Λֻ͚Ε͹ઌ಄߲͕ফ͑Δɻ 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  X  a2  =  XY − 1 Y2 − 1 X2Y + XY2 + Y2 X2Y − X XY2 + X Ҿ͘ɻ 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  X  a2  =  XY − 1 Y2 − 1 X2Y + XY2 + Y2 X2Y − X XY2 + X + Y2 ্͔ΒҰͭ߱Ζͯ͠ɺ߱ႈͷॱʹฒ΂Δ 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  X  +  a2  =  XY − 1 Y2 − 1 X2Y + XY2 + Y2 X2Y − X XY2 + X + Y2 ઌ಄߲ʹ஫໨ɻ XY − 1 ͰׂΕΔɻ 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  X  +  Y a2  =  XY − 1 Y2 − 1 X2Y + XY2 + Y2 X2Y − X XY2 + X + Y2 XY2 − Y X + Y2 + Y ઌఔͱಉ༷ʹܭࢉɻ 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  X  +  Y a2  =  XY − 1 Y2 − 1 X2Y + XY2 + Y2 X2Y − X XY2 + X + Y2 XY2 − Y X + Y2 − Y ઌ಄߲ʹ஫໨ɻXY, Y2 ͲͪΒͰ΋ׂΕͳ͍ɻ 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  X  +  Y a2  =  1 XY − 1 Y2 − 1 X2Y + XY2 + Y2 X2Y − X XY2 + X + Y2 XY2 − Y X + Y2 − Y X ׂΕͳ͍ͷͰ X ͸༨Γͱͯ͠ԣʹΑ͚Δ Y2 − Y 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  X  +  Y a2  =  1 XY − 1 Y2 − 1 X2Y + XY2 + Y2 X2Y − X XY2 + X + Y2 XY2 − Y X + Y2 − Y X Y2 − Y ઌ಄߲ʹ஫໨ɻ XY ͸ແཧͩ 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  X  +  Y a2  =  1 XY − 1 Y2 − 1 X2Y + XY2 + Y2 X2Y − X XY2 + X + Y2 XY2 − Y X + Y2 − Y X Y2 ͰׂΕΔͷͰɺׂΔɻ Y2 − Y Y2 − 1 Y + 1 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  X  +  Y a2  =  1 XY − 1 Y2 − 1 X2Y + XY2 + Y2 X2Y − X XY2 + X + Y2 XY2 − Y X + Y2 − Y X Y2 − Y Y2 − 1 Y + 1 Y 1 0 1 Y, 1 ڞʹͲͷ߲Ͱ΋ׂΕͳ͍ͷͰԣʹΑ͚Δ 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  X  +  Y a2  =  1 XY − 1 Y2 − 1 X2Y + XY2 + Y2 X2Y − X XY2 + X + Y2 XY2 − Y X + Y2 − Y X Y2 − Y Y2 − 1 Y + 1 Y 1 0 1 r = X + Y + 1 Α͚ͨͷΛ଍ͯ͠ɺ༨Γ r ͕ग़Δ 13೥5݄4೔౔༵೔

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࣮ྫɿଟม਺ͷׂΓࢉ a1  =  X  +  Y a2  =  1 XY − 1 Y2 − 1 X2Y + XY2 + Y2 X2Y − X XY2 + X + Y2 XY2 − Y X + Y2 − Y X Y2 − Y Y2 − 1 Y + 1 Y 1 0 1 r = X + Y + 1 X2Y + XY2 + Y2 = (XY − 1) (X+Y) + (Y2 − 1) ⋅1 + X + Y + 1 13೥5݄4೔౔༵೔

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ଟม਺ׂΓࢉ ͜ΕͰ͍͍ͷͰ͸……ʂʁ ͱ͜Ζ͕ɺ ͋·Γ خ͘͠ͳ͍͜ͱ͕…… 13೥5݄4೔౔༵೔

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͋·Γ ͷ໰୊ a1  =  a2  =  Y2 − 1 XY − 1 X2Y + XY2 + Y2 ׂΔॱ൪Λม͑ɺY2 − 1, XY − 1Ͱ X2Y + XY2 + Y2 ΛׂΔɻ 13೥5݄4೔౔༵೔

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༨Γͷ໰୊ a1  =  X + 1 a2  = X  Y2 − 1 XY − 1 X2Y + XY2 + Y2 X2Y − X XY2 + X + Y2 XY2 − X 2X + Y2 2X Y2 Y2 − 1 1 0 1 r = 2X + 1 X2Y + XY2 + Y2 = (XY − 1) X + (Y2 − 1)(X + 1) + 2X + 1 13೥5݄4೔౔༵೔

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༨Γ͕ҧ͏ʂ 13೥5݄4೔౔༵೔

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༨Γ͕ҧ͏ʂ ଟ߲ࣜͷׂΓࢉ͸ ׂΔॱ൪Ͱ༨Γ͕มΘΔʂ 13೥5݄4೔౔༵೔

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༨Γ͕ҧ͏ʂ ଟ߲ࣜͷׂΓࢉ͸ ׂΔॱ൪Ͱ༨Γ͕มΘΔʂ ୯ʹʮׂͬͯ༨Γθϩ㱺ΠσΞϧʹೖΔʂʯͱ ͸ग़དྷͳ͍ʂ 13೥5݄4೔౔༵೔

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༨Γ͕ҧ͏ʂ ଟ߲ࣜͷׂΓࢉ͸ ׂΔॱ൪Ͱ༨Γ͕มΘΔʂ ୯ʹʮׂͬͯ༨Γθϩ㱺ΠσΞϧʹೖΔʂʯͱ ͸ग़དྷͳ͍ʂ ༩͑ΒΕͨΠσΞϧʹରͯ͠ɺׂΓࢉͷ༨Γ͕ ॱ൪ʹґΒͳ͍Α͏ͳੜ੒ݩΛऔΕͳ͍͔ʁ 13೥5݄4೔౔༵೔

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༨Γ͕ҧ͏ʂ ଟ߲ࣜͷׂΓࢉ͸ ׂΔॱ൪Ͱ༨Γ͕มΘΔʂ ୯ʹʮׂͬͯ༨Γθϩ㱺ΠσΞϧʹೖΔʂʯͱ ͸ग़དྷͳ͍ʂ ༩͑ΒΕͨΠσΞϧʹରͯ͠ɺׂΓࢉͷ༨Γ͕ ॱ൪ʹґΒͳ͍Α͏ͳੜ੒ݩΛऔΕͳ͍͔ʁ ➡ Gröbner جఈʂ 13೥5݄4೔౔༵೔

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Gröbner جఈ ୯߲ࣜॱং > ΛҰͭݻఆ͢Δɻଟ߲ࣜ؀ͷΠσΞ ϧ I ⊆ k[] ʹ͍ͭͯɺI ͷ༗ݶ෦෼ू߹ G = {g1, …, gn } ⊆ I ͕࣍ͷ৚݅Λຬͨ͢ͱ͖ɺG ͸ > ʹؔ͢ Δ I ͷ Gröbner جఈ Ͱ͋Δͱݴ͏ɻ ⟨LT(I)⟩ = ⟨LT(g1 ), …, LT(gn )⟩ (i.e. ⟨LT(f) ∣ f ∈ I ⟩ = ⟨LT(g1 ), …, LT(gn )⟩) ఆٛ 13೥5݄4೔౔༵೔

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Gröbner جఈͷੑ࣭ ఆཧ 13೥5݄4೔౔༵೔

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Gröbner جఈͷੑ࣭ ೚ҙͷଟ߲ࣜ؀ΠσΞϧʹର͠ Gröbner جఈ͕ଘࡏ͢Δ ఆཧ 13೥5݄4೔౔༵೔

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Gröbner جఈͷੑ࣭ ೚ҙͷଟ߲ࣜ؀ΠσΞϧʹର͠ Gröbner جఈ͕ଘࡏ͢Δ ΠσΞϧ I ͷ Gröbner جఈ G = {g1 , …, gk } ͸ I Λੜ੒ ͢ΔɻଈͪɺI = ⟨g1 , …, gk ⟩ ఆཧ 13೥5݄4೔౔༵೔

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Gröbner جఈͷੑ࣭ ೚ҙͷଟ߲ࣜ؀ΠσΞϧʹର͠ Gröbner جఈ͕ଘࡏ͢Δ ΠσΞϧ I ͷ Gröbner جఈ G = {g1 , …, gk } ͸ I Λੜ੒ ͢ΔɻଈͪɺI = ⟨g1 , …, gk ⟩ Gröbner جఈʹΑΔׂΓࢉͷ༨Γ͸ॱ൪ʹґΒͳ͍ɻ ఆཧ 13೥5݄4೔౔༵೔

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Gröbner جఈͷੑ࣭ ೚ҙͷଟ߲ࣜ؀ΠσΞϧʹର͠ Gröbner جఈ͕ଘࡏ͢Δ ΠσΞϧ I ͷ Gröbner جఈ G = {g1 , …, gk } ͸ I Λੜ੒ ͢ΔɻଈͪɺI = ⟨g1 , …, gk ⟩ Gröbner جఈʹΑΔׂΓࢉͷ༨Γ͸ॱ൪ʹґΒͳ͍ɻ Gröbner جఈͰׂͬͨ༨Γ͕θϩͰ͋Δࣄͱ I ʹଐ͢Δ ͜ͱ͸ಉ஋ɿ r = f ̅G = 0 ⟺ f ∈ I ఆཧ 13೥5݄4೔౔༵೔

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·ͱΊ Gröbner جఈ͸ମ্ଟ߲ࣜ؀ͷΠσΞϧͷྑ͍ੑ ࣭Λ࣋ͬͨੜ੒ݩ ୯߲ࣜॱংΛܾΊΔͱܭࢉग़དྷΔ ׂΓࢉͷ༨Γ͕ॱ൪ʹґΒͳ͍ ΠσΞϧॴଐ໰୊ʹ࢖͑Δ ଞʹ΋Ԡ༻ଟ਺ 13೥5݄4೔౔༵೔

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͜͜·ͰͰ࣭໰ʁ 13೥5݄4೔౔༵೔

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Table of Contents Gröbner جఈͷ؆୯ͳಋೖ Gröbner جఈͷܭࢉ๏ Buchberger ΞϧΰϦζϜ ࠷దԽख๏ɿsyzygy, sugar strategy Haskell ͷܕϨϕϧϓϩάϥϛϯάͷݱࡏ 13೥5݄4೔౔༵೔

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Gröbner جఈͷܭࢉ๏ 13೥5݄4೔౔༵೔

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Buchberger ΞϧΰϦζϜ 13೥5݄4೔౔༵೔

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Buchberger ΞϧΰϦζϜ Gröbner جఈ͕ศརͦ͏ͳͷ͸Θ͔ͬͨ 13೥5݄4೔౔༵೔

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Buchberger ΞϧΰϦζϜ Gröbner جఈ͕ศརͦ͏ͳͷ͸Θ͔ͬͨ ۩ମతͳܭࢉ๏ɿBuchberger ΞϧΰϦζϜ 13೥5݄4೔౔༵೔

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Buchberger ΞϧΰϦζϜ Gröbner جఈ͕ศརͦ͏ͳͷ͸Θ͔ͬͨ ۩ମతͳܭࢉ๏ɿBuchberger ΞϧΰϦζϜ ४උతͳఆٛɿf, g ∈ k[] ͷ S-ଟ߲ࣜ 13೥5݄4೔౔༵೔

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Buchberger ΞϧΰϦζϜ Gröbner جఈ͕ศརͦ͏ͳͷ͸Θ͔ͬͨ ۩ମతͳܭࢉ๏ɿBuchberger ΞϧΰϦζϜ ४උతͳఆٛɿf, g ∈ k[] ͷ S-ଟ߲ࣜ S(f, g) := ɹɹ ɹɹ f − ɹɹ  ɹɹg LCM(LT(f), LT(g)) LT(f) LCM(LT(f), LT(g)) LT(g) 13೥5݄4೔౔༵೔

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Buchberger ΞϧΰϦζϜ Gröbner جఈ͕ศརͦ͏ͳͷ͸Θ͔ͬͨ ۩ମతͳܭࢉ๏ɿBuchberger ΞϧΰϦζϜ ४උతͳఆٛɿf, g ∈ k[] ͷ S-ଟ߲ࣜ S(f, g) := ɹɹ ɹɹ f − ɹɹ  ɹɹg f, g ͷઌ಄߲ಉ࢜Λଧফ͠߹Θͤͨ෺ LCM(LT(f), LT(g)) LT(f) LCM(LT(f), LT(g)) LT(g) 13೥5݄4೔౔༵೔

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Buchberger ΞϧΰϦζϜ Gröbner جఈ͕ศརͦ͏ͳͷ͸Θ͔ͬͨ ۩ମతͳܭࢉ๏ɿBuchberger ΞϧΰϦζϜ ४උతͳఆٛɿf, g ∈ k[] ͷ S-ଟ߲ࣜ S(f, g) := ɹɹ ɹɹ f − ɹɹ  ɹɹg f, g ͷઌ಄߲ಉ࢜Λଧফ͠߹Θͤͨ෺ ͨͩ͠ LCM((a1 , …, an ), (b1 , …, bn ))  := (max{a1 , b1 }, …, max{an , bn }) LCM(LT(f), LT(g)) LT(f) LCM(LT(f), LT(g)) LT(g) 13೥5݄4೔౔༵೔

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Buchberger ൑ఆ๏ I ⊆ k[]ɿΠσΞϧɹɹ >ɿ୯߲ࣜॱং G = {g1 , …, gk } ⊆ IɿI ͷੜ੒ݩ G’ = (g1 , …, gk )ɿG ͷݩΛద౰ʹฒ΂ͨ΋ͷ ͜ͷͱ͖ҎԼͷೋͭ͸ಉ஋ɿ 1. G ͸ > ʹؔ͢Δ I ͷ Gröbner جఈ 2.S(gi , gk )G’ = 0ɹ(∀ i < j) ఆཧ 13೥5݄4೔౔༵೔

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Buchberger ΞϧΰϦζϜ ೖྗɿG = (f1 , …, fs ) R ← {S(p, q)G’ ∣ p ≠ q ∈ G} \ {0} G ← (G, R) R ≠ ∅ ग़ྗɿG R = ∅ 13೥5݄4೔౔༵೔

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؆୯ͳ࣮૷ buchberger :: [Polynomial k ord n] → [Polynomial k ord n] buchberger ideal = fst $ until (null ∘ snd) loop (ideal, calc ideal) where loop (ggs, acc) = let cur = nub $ ggs ++ acc in (cur, calc cur) calc acc = [ q | f ← acc, g ← acc, f ≠ g , let q = sPolynomial f g `mod` acc , q ≠ zero ] 13೥5݄4೔౔༵೔

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ܭࢉྫ 13೥5݄4೔౔༵೔

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ܭࢉྫ I = ⟨x2y − 1, x3 − y2 − x⟩ ͷ Gröbner جఈ 13೥5݄4೔౔༵೔

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ܭࢉྫ I = ⟨x2y − 1, x3 − y2 − x⟩ ͷ Gröbner جఈ Grevlex Ͱܭࢉ G = {x2y − 1, x3 − y2 − x, y3 + xy − x,  − y3 − xy + x} 13೥5݄4೔౔༵೔

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ܭࢉྫ I = ⟨x2y − 1, x3 − y2 − x⟩ ͷ Gröbner جఈ Grevlex Ͱܭࢉ G = {x2y − 1, x3 − y2 − x, y3 + xy − x,  − y3 − xy + x} ׂΔॱ൪Ͱ౴͕͑มΘΒͳ͍͜ͱ͸ɺ·͋ɺ֬ ͔Ίͱ͍͍ͯͩ͘͞ɻ 13೥5݄4೔౔༵೔

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lex Ͱܭࢉ͢Δͱʁ 13೥5݄4೔౔༵೔

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lex Ͱܭࢉ͢Δͱʁ G = {x2y − 1, x3 − y2 − x, y3 + xy − x,  − y3 − xy + x, y5 + y4 +  y3 + x2 − x − 1,  − y5 − y4 − y3 − x2 + x + 1,  − y6 − y5 − y4 −  y3 + x + y − 1, y7 + 2y6 + 2y5 + 2y4 + 2y3 − y2 − 2x + 1, y6 +  y5 + y4 + y3 − x − y + 1,  − y7 − 2y6 − 2y5 − 2y4 − 2y3 + y2 +  2x − 1,  − y9 − 2y8 − 3y7 + y4 − 4y + 3,  − 1 / 2y10 − 3 / 2y9 −  5 / 2y8 − 7 / 2y7 + 1 / 2y5 + 1 / 2y4 − 9 / 2y + 7 / 2, y7 − y2 + 2y  − 1, 1 / 2y8 + y7 − 1 / 2y3 + 3 / 2y − 1,  − y7 + y2 − 2y + 1,  −  y8 − 2y7 + y3 − 3y + 2,  − 1 / 2y9 − 3 / 2y8 − 5 / 2y7 + 1 / 2y4 +  1 / 2y3 − 7 / 2y + 5 / 2, y9 + 2y8 + 3y7 − y4 + 4y − 3, y8 + 2y7 −  y3 + 3y − 2, 1 / 2y7 − 1 / 2y2 + y − 1 / 2, 1 / 2y10 + 3 / 2y9 + 5 /  2y8 + 7 / 2y7 − 1 / 2y5 − 1 / 2y4 + 9 / 2y − 7 / 2,  − 1 / 2y8 − y7  + 1 / 2y3 − 3 / 2y + 1, 1 / 2y9 + 3 / 2y8 + 5 / 2y7 − 1 / 2y4 − 1 /  2y3 + 7 / 2y − 5 / 2,  − 1 / 2y7 + 1 / 2y2 − y + 1 / 2} 13೥5݄4೔౔༵೔

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Ͱ͔ͬʂ 13೥5݄4೔౔༵೔

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ޮ཰ͷ໰୊ Grevlex ͕Ұ൪଎ͯ͘جఈ΋៉ྷͳ΋ͷ͕ٻ·Γ ΍͍͢ͱ͞ΕΔ lex ͸جఈ΋ෳࡶʹͳΔ͕͔͔࣌ؒ͠Δ ෺ʹΑͬͯ͸ Grevlex ͷํ͕஗͍͜ͱ΋͋Δ ͕ɺ΄΅େ఍͸ Grevlex ͕ѹ౗తʹ଎͍ Ԡ༻෼໺ʹΑͬͯ͸ grevlex Ͱ͸ͳ͘ lex Λ࢖͏ 13೥5݄4೔౔༵೔

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ۃখɾඃ໿Gröbnerجఈ 13೥5݄4೔౔༵೔

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ۃখɾඃ໿Gröbnerجఈ Buchberger ΞϧΰϦζϜͰٻΊͨجఈʹ͸ແବ͕͋Δ 13೥5݄4೔౔༵೔

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ۃখɾඃ໿Gröbnerجఈ Buchberger ΞϧΰϦζϜͰٻΊͨجఈʹ͸ແବ͕͋Δ ۃখɾඃ໿ Gröbner جఈʂ 13೥5݄4೔౔༵೔

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ۃখɾඃ໿Gröbnerجఈ Buchberger ΞϧΰϦζϜͰٻΊͨجఈʹ͸ແବ͕͋Δ ۃখɾඃ໿ Gröbner جఈʂ Gröbnerجఈ G ͕ۃখ ⇔ ೚ҙͷ i ʹ͍ͭͯ LC(gi ) = 1, LT(gi ) ͸Ͳͷ LT(gj ) (j ≠ i) Ͱ΋ׂΕͳ͍ 13೥5݄4೔౔༵೔

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ۃখɾඃ໿Gröbnerجఈ Buchberger ΞϧΰϦζϜͰٻΊͨجఈʹ͸ແବ͕͋Δ ۃখɾඃ໿ Gröbner جఈʂ Gröbnerجఈ G ͕ۃখ ⇔ ೚ҙͷ i ʹ͍ͭͯ LC(gi ) = 1, LT(gi ) ͸Ͳͷ LT(gj ) (j ≠ i) Ͱ΋ׂΕͳ͍ G ͕ඃ໿  ⇔ ೚ҙͷ i ʹ͍ͭͯ LC(gi ) = 1 ͔ͭ  gi ͷͲͷ߲΋ LT(gj ) (j ≠ i) Ͱ΋ׂΕͳ͍ 13೥5݄4೔౔༵೔

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ඃ໿ Gröbner جఈͷҰҙੑ 13೥5݄4೔౔༵೔

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ඃ໿ Gröbner جఈͷҰҙੑ Gröbner جఈ͸ෳ਺ଘࡏ͠͏Δ 13೥5݄4೔౔༵೔

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ඃ໿ Gröbner جఈͷҰҙੑ Gröbner جఈ͸ෳ਺ଘࡏ͠͏Δ buchberger ͰٻΊͨ෺ɺͦ ͷۃখԽ……etc 13೥5݄4೔౔༵೔

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ඃ໿ Gröbner جఈͷҰҙੑ Gröbner جఈ͸ෳ਺ଘࡏ͠͏Δ buchberger ͰٻΊͨ෺ɺͦ ͷۃখԽ……etc ͦΕΒ͸ඃ໿Խ͢Ε͹Ұக͢ Δʂʢ୯߲ࣜॱং͕ಉ͡ͳΒʣ I ⊆ k[]ɿΠσΞϧ >ɿ୯߲ࣜॱং I ͷ > ʹ͍ͭͯͷඃ ໿ Gröbner جఈ͸Ұ ҙʹఆ·Δ ఆཧ 13೥5݄4೔౔༵೔

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ඃ໿ Gröbner جఈͷҰҙੑ Gröbner جఈ͸ෳ਺ଘࡏ͠͏Δ buchberger ͰٻΊͨ෺ɺͦ ͷۃখԽ……etc ͦΕΒ͸ඃ໿Խ͢Ε͹Ұக͢ Δʂʢ୯߲ࣜॱং͕ಉ͡ͳΒʣ ΠσΞϧ͕౳͍͔͠Ͳ͏͔Λɺ ඃ໿ Gröbner جఈͷੜ੒ݩ͕ ౳͍͔͠Ͱ൑ఆग़དྷΔʂ I ⊆ k[]ɿΠσΞϧ >ɿ୯߲ࣜॱং I ͷ > ʹ͍ͭͯͷඃ ໿ Gröbner جఈ͸Ұ ҙʹఆ·Δ ఆཧ 13೥5݄4೔౔༵೔

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ඃ໿Խͷखଓ 13೥5݄4೔౔༵೔

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ඃ໿Խͷखଓ ·ͣۃখԽΛߦ͏ 1. G ͷ֤ݩΛ࠷ߴ࣍܎਺Ͱׂͬͯ LC(p) = 1 2. (2) ͷ৚݅Λຬͨ͞ͳ͍෺ΛऔΓআ͘ 13೥5݄4೔౔༵೔

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ඃ໿Խͷखଓ ·ͣۃখԽΛߦ͏ 1. G ͷ֤ݩΛ࠷ߴ࣍܎਺Ͱׂͬͯ LC(p) = 1 2. (2) ͷ৚݅Λຬͨ͞ͳ͍෺ΛऔΓআ͘ ͦΕΛඃ໿Խ͢Δ 1. p ∈ G ΛऔΓɺp’ = p̅G \ {p}, G’ = G \ {p} ∪ {p’} ͱ͢Δ 2. q ∈ G’ \ {p’} ʹ͍ͭͯಉ༷ͷࣄΛ͢ΔɻҎԼಉจɻ 13೥5݄4೔౔༵೔

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ܭࢉྫ 13೥5݄4೔౔༵೔

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ܭࢉྫ ઌఔͷΠσΞϧʹ͍ͭͯ 13೥5݄4೔౔༵೔

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ܭࢉྫ ઌఔͷΠσΞϧʹ͍ͭͯ Grevlex ʹؔ͢Δඃ໿جఈ G = {y3 + xy − x, x2y − 1, x3 − y2 − x} 13೥5݄4೔౔༵೔

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ܭࢉྫ ઌఔͷΠσΞϧʹ͍ͭͯ Grevlex ʹؔ͢Δඃ໿جఈ G = {y3 + xy − x, x2y − 1, x3 − y2 − x} Lex ʹؔ͢Δඃ໿جఈ G = {y7 − y2 + 2y − 1,  − y6 − y5 − y4 − y3 + x  + y − 1} 13೥5݄4೔౔༵೔

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ܭࢉྫ ઌఔͷΠσΞϧʹ͍ͭͯ Grevlex ʹؔ͢Δඃ໿جఈ G = {y3 + xy − x, x2y − 1, x3 − y2 − x} Lex ʹؔ͢Δඃ໿جఈ G = {y7 − y2 + 2y − 1,  − y6 − y5 − y4 − y3 + x  + y − 1} ͏Θͬ…φΠʔϒͳجఈɺແବଟ͗͢…ʁ 13೥5݄4೔౔༵೔

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ܭࢉྫ ઌఔͷΠσΞϧʹ͍ͭͯ Grevlex ʹؔ͢Δඃ໿جఈ G = {y3 + xy − x, x2y − 1, x3 − y2 − x} Lex ʹؔ͢Δඃ໿جఈ G = {y7 − y2 + 2y − 1,  − y6 − y5 − y4 − y3 + x  + y − 1} ͏Θͬ…φΠʔϒͳجఈɺແବଟ͗͢…ʁ ࠷ॳ͔Βඃ໿άϨϒφʔجఈΛ࢈ΉΞϧΰϦζ Ϝ΋͋Δ 13೥5݄4೔౔༵೔

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࣮૷ྫ దٓΞοϓσʔτ͢Δඞཁ͕͋ΔͷͰ ST Ͱָͯ͠ॻ͍ͨ ඃ໿Խͷ࣮૷΋ಉ༷ minimizeGroebnerBasis :: (Field k, IsPolynomial k n, IsMonomialOrder order) ⇒ [OrderedPolynomial k ord n] → [OrderedPolynomial k ord n] minimizeGroebnerBasis bs = runST $ do left ← newSTRef bs right ← newSTRef [] whileM_ (not . null <$> readSTRef left) $ do f : xs ← readSTRef left writeSTRef left xs ys ← readSTRef right if any (λg → leadingMonomial g `divs` leadingMonomial f) xs ∨ any (λg → leadingMonomial g `divs` leadingMonomial f) ys then writeSTRef right ys else writeSTRef right (monoize f : ys) readSTRef right 13೥5݄4೔౔༵೔

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Table of Contents Gröbner جఈͷ؆୯ͳಋೖ Gröbner جఈͷܭࢉ๏ Buchberger ΞϧΰϦζϜ ࠷దԽख๏ɿsyzygy, sugar strategy Haskell ͷܕϨϕϧϓϩάϥϛϯάͷݱࡏ 13೥5݄4೔౔༵೔

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࠷దԽख๏ φΠʔϒͳ࣮૷ͩͱ஗͍ʂ ୅දతͳ࣮૷Ͱ͋Δ SINGULAR ͷઍഒ͘Β͍ ஗͍ʢਪఆʣ Կͱ͔ඦഒ͘Β͍ʹ͢Δํ๏ ࢦ਺ΦʔμʔͳͷͰʙഒ͍ͬͯ͏දݱ͸͓ ͔͍͠ͷ͚ͩͲɾɾɾ 13೥5݄4೔౔༵೔

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঺հ͢Δख๏ θϩ؆໿ؔ܎ syzygy ͷجఈ sugar strategy 13೥5݄4೔౔༵೔

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঺հ͢Δख๏ θϩ؆໿ؔ܎ syzygy ͷجఈ sugar strategy 13೥5݄4೔౔༵೔

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θϩ؆໿ؔ܎ 13೥5݄4೔౔༵೔

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θϩ؆໿ؔ܎ Buchberger ͸Կނ஗͍ʁ 13೥5݄4೔౔༵೔

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θϩ؆໿ؔ܎ Buchberger ͸Կނ஗͍ʁ Buchberger ͸ׂΓࢉͷ͔ͨ·Γ 13೥5݄4೔౔༵೔

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θϩ؆໿ؔ܎ Buchberger ͸Կނ஗͍ʁ Buchberger ͸ׂΓࢉͷ͔ͨ·Γ lex ΑΓ grevlex ͕଎͍ͷ͸ɺͦͬͪͷ΄ ͏ׂ͕Γࢉͷεςοϓ͕খ͍͔͞Β 13೥5݄4೔౔༵೔

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θϩ؆໿ؔ܎ Buchberger ͸Կނ஗͍ʁ Buchberger ͸ׂΓࢉͷ͔ͨ·Γ lex ΑΓ grevlex ͕଎͍ͷ͸ɺͦͬͪͷ΄ ͏ׂ͕Γࢉͷεςοϓ͕খ͍͔͞Β ׂΓࢉͷճ਺Λ΋ͬͱݮΒͤͳ͍ʁ 13೥5݄4೔౔༵೔

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θϩ؆໿ؔ܎ ʮׂͬͯ༨ΓθϩʯΛҰൠԽͨ͠ͷ͕࣍ 13೥5݄4೔౔༵೔

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θϩ؆໿ؔ܎ ʮׂͬͯ༨ΓθϩʯΛҰൠԽͨ͠ͷ͕࣍ ଟ߲ࣜ f ∈ k[] ͕ G = {g1 , …, gk } ⊆ k[] Λ ๏ͱͯ͠θϩʹ؆໿ʢه߸ɿ f →G 0ʣ͞ΕΔ  ⇔ f = a1 g1  + ⋯ + ak gk (ai  ∈ k[]) ͔ͭ [ai gi  = 0 ·ͨ͸ deg(f) ≥ deg(ai gi )] ୠ͠ɺdeg(f) ͸ f ͷଟॏ࣍਺ deg(f) = LM(f) ఆٛ 13೥5݄4೔౔༵೔

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վྑ Buchberger ൑ఆ๏ ࣮͸ Buchberger ൑ఆ๏͸θϩ؆໿Ͱे෼ʂ 13೥5݄4೔౔༵೔

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վྑ Buchberger ൑ఆ๏ ࣮͸ Buchberger ൑ఆ๏͸θϩ؆໿Ͱे෼ʂ I ⊆ k[]ɿΠσΞϧɹɹ >ɿ୯߲ࣜॱং G = {g1 , …, gk } ⊆ IɿI ͷੜ੒ݩ ͜ͷͱ͖ҎԼͷೋͭ͸ಉ஋ɿ 1. G ͸ > ʹؔ͢Δ I ͷ Gröbner جఈ 2. S(gi , gj )  →G   0ɹ(∀ i < j) ఆཧ 13೥5݄4೔౔༵೔

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۩ମతʹ͍ͭθϩ؆໿ʁ I ⊆ k[]ɿΠσΞϧɹɹ >ɿ୯߲ࣜॱং G = {g1 , …, gk } ⊆ IɿI ͷੜ੒ݩ ໋୊ 13೥5݄4೔౔༵೔

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۩ମతʹ͍ͭθϩ؆໿ʁ I ⊆ k[]ɿΠσΞϧɹɹ >ɿ୯߲ࣜॱং G = {g1 , …, gk } ⊆ IɿI ͷੜ੒ݩ S̅(f̅, ̅g̅)̅G’ = 0 ⇒ S(f, g) →G 0 ໋୊ 13೥5݄4೔౔༵೔

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۩ମతʹ͍ͭθϩ؆໿ʁ I ⊆ k[]ɿΠσΞϧɹɹ >ɿ୯߲ࣜॱং G = {g1 , …, gk } ⊆ IɿI ͷੜ੒ݩ S̅(f̅, ̅g̅)̅G’ = 0 ⇒ S(f, g) →G 0 LCM(LM(f), LM(g)) = LM(f) LM(g) ⇒ S(f, g)  →G 0 ʢf, g ͕ ޓ͍ʹૉ ͳΒ S-ଟ߲ࣜ͸θϩʹ؆໿͞ΕΔʣ ໋୊ 13೥5݄4೔౔༵೔

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ޓ͍ʹૉ൑ఆ ೋ൪໨ͷ৚͕݅࢖͑Δʂ x3 ͱ z4 ͸ޓ͍ʹૉͳͷͰ S(x3 + y, z4) →G 0 ͱͳΔ ɹɹɹ S(x3 + y, z4) = yz4 ɹɹɹ∴  S(x3 + y, z4)G = y ≠ 0 ࠓ·Ͱͷ൑ఆ๏ͩͱແବʹׂΓࢉΛ͢Δʂ ͲΕ͘Β͍վળʁ 13೥5݄4೔౔༵೔

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Benchmark simple prime I1  = ⟨x2 + y2 + z2 − 1, x2 + y2 + z2 − 2x, 2x − 3y − z⟩ I2  = ⟨x2y − 2xy − 4z − 1, z − y2, x3 − 4zy⟩ I3  = ⟨2S − ay, b2 − (x2 + y2), c2 − ((a − x)2 + y2)⟩" I4  = ⟨z5 + y4 + x3 − 1, z3 + y3 + x2 − 1⟩ 13೥5݄4೔౔༵೔

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Benchmark lex01 grlex01 grevlex01 lex03 grlex03 grevlex03 0ms 22.5ms 45ms 67.5ms 90ms simple prime grlex02 grevlex02 0.9s 1s 1.1s 1.2s 1.3s grlex04 grevlex04 0μs 150μs 300μs 450μs 600μs I1  = ⟨x2 + y2 + z2 − 1, x2 + y2 + z2 − 2x, 2x − 3y − z⟩ I2  = ⟨x2y − 2xy − 4z − 1, z − y2, x3 − 4zy⟩ I3  = ⟨2S − ay, b2 − (x2 + y2), c2 − ((a − x)2 + y2)⟩" I4  = ⟨z5 + y4 + x3 − 1, z3 + y3 + x2 − 1⟩ 13೥5݄4೔౔༵೔

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঺հ͢Δख๏ θϩ؆໿ؔ܎ syzygy ͷجఈ sugar strategy 13೥5݄4೔౔༵೔

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syzygy ͷجఈ ͦ΋ͦ΋ S-ଟ߲ࣜͱ͸Կͳͷ͔ʁ ࣮͸ syzygy ͱݺ͹ΕΔ΋ͷͷجఈʂ Gröbner جఈͷجఈͱ͸ҙຯ͕एׯҧ͏ “syzygy ͷ΋ͱ” ͱࢥ͑͹Α͍ 13೥5݄4೔౔༵೔

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syzygy ఆٛ 13೥5݄4೔౔༵೔

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syzygy (h1 , …, hs ) ∈ k[]s ͕ F = (f1 , …, fs ) ⊆ k[]s ͷ ઌ಄߲ͷ syzygy  :⇔ h1 LT(f1 ) + ⋯ + hs LT(fs ) = 0 F ͷઌ಄߲ syzygy ͷશମΛ S(F) ͱॻ͘ɻ ఆٛ 13೥5݄4೔౔༵೔

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syzygy (h1 , …, hs ) ∈ k[]s ͕ F = (f1 , …, fs ) ⊆ k[]s ͷ ઌ಄߲ͷ syzygy  :⇔ h1 LT(f1 ) + ⋯ + hs LT(fs ) = 0 F ͷઌ಄߲ syzygy ͷશମΛ S(F) ͱॻ͘ɻ T ⊆ S(F) ͕ S(F) ͷجఈ  ⇔ S(F) ͷͲͷݩ΋ T ͷݩͷ༗ ݶݸͷ࿨Ͱॻ͚Δɻ ఆٛ 13೥5݄4೔౔༵೔

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syzygy (h1 , …, hs ) ∈ k[]s ͕ F = (f1 , …, fs ) ⊆ k[]s ͷ ઌ಄߲ͷ syzygy  :⇔ h1 LT(f1 ) + ⋯ + hs LT(fs ) = 0 F ͷઌ಄߲ syzygy ͷશମΛ S(F) ͱॻ͘ɻ T ⊆ S(F) ͕ S(F) ͷجఈ  ⇔ S(F) ͷͲͷݩ΋ T ͷݩͷ༗ ݶݸͷ࿨Ͱॻ͚Δɻ ei  = (0, …, 0, 1, 0, …, 0) (i ൪໨͚ͩ 1) ͱॻ͘͜ͱʹ͢ Δɻ͜ͷͱ͖ ͱஔ͘ͱɺ S(fi , fj ) = Sij  ⋅ F ʢ಺ੵʣͱͳΔɻ LCM(LT(fi ), LT(fj )) LT(fi) LCM(LT(fi ), LT(fj )) LT(fj) Sij  :=  ei −   ej ఆٛ 13೥5݄4೔౔༵೔

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syzygy ͷجఈͱθϩ؆໿ 13೥5݄4೔౔༵೔

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syzygy ͷجఈͱθϩ؆໿ I ⊆ k[]ɿΠσΞϧɹɹ >ɿ୯߲ࣜॱং G = {g1 , …, gk } ⊆ IɿI ͷੜ੒ݩ S ⊆ S(G)ɿG ͷઌ಄߲ syzygy ͷجఈ ͜ͷͱ͖ҎԼͷೋͭ͸ಉ஋ɿ 1. G ͸ > ʹؔ͢Δ I ͷ Gröbner جఈ 2.೚ҙͷ T ∈ S ʹର͠ɺ T ⋅ G = ∑ k i = 1 ti gi  →G   0ɹ(∀ i < j) ఆཧ 13೥5݄4೔౔༵೔

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S-ଟ߲ࣜͱ syzygy 13೥5݄4೔౔༵೔

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S-ଟ߲ࣜͱ syzygy ࣮͸ Sij ͷશମ͸ S(G) ͷجఈ 13೥5݄4೔౔༵೔

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S-ଟ߲ࣜͱ syzygy ࣮͸ Sij ͷશମ͸ S(G) ͷجఈ ༨෼ͳجఈ΋͍ࠞͬͯ͡Δ F = (x2y2 + z, xy2 − y, x2y + yz) S12  = (1,  -x, 0) S13  = (1, 0,  -y) S23  = (0, x,  -y) S13  − S12  = (0, x,  -y) = S23 ∴ S23 ͸ཁΒͳ͍ࢠ 13೥5݄4೔౔༵೔

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S-ଟ߲ࣜͱ syzygy ࣮͸ Sij ͷશମ͸ S(G) ͷجఈ ༨෼ͳجఈ΋͍ࠞͬͯ͡Δ F = (x2y2 + z, xy2 − y, x2y + yz) S12  = (1,  -x, 0) S13  = (1, 0,  -y) S23  = (0, x,  -y) S13  − S12  = (0, x,  -y) = S23 ∴ S23 ͸ཁΒͳ͍ࢠ ༨෼ͳجఈΛ༧Ί൑ఆग़དྷͳ͍͔ʁ 13೥5݄4೔౔༵೔

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ແବͳ syzygy جఈͷ൑ఆ 13೥5݄4೔౔༵೔

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ແବͳ syzygy جఈͷ൑ఆ G = (g1 , …, gs ) gi , gj , gk  ∈ G ⊆ {Sij ∣ 1 ≤ j < k ≤ s}ɿS(G) ͷجఈ LT(gk ) ∣ LCM(LT(gi ), LT(gj )) ͔ͭ Sik , Sjk  ∈  ⇒  S \ {Sij } ΋ S(G) ͷجఈ ໋୊ 13೥5݄4೔౔༵೔

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ແବͳ syzygy جఈͷ൑ఆ G = (g1 , …, gs ) gi , gj , gk  ∈ G ⊆ {Sij ∣ 1 ≤ j < k ≤ s}ɿS(G) ͷجఈ LT(gk ) ∣ LCM(LT(gi ), LT(gj )) ͔ͭ Sik , Sjk  ∈  ⇒  S \ {Sij } ΋ S(G) ͷجఈ ໋୊ ͜ΕͰׂΓࢉͷճ਺Λେ෯ʹݮΒͤΔʂ 13೥5݄4೔౔༵೔

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վྑ൛ Buchbergerɾվ 13೥5݄4೔౔༵೔

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վྑ൛ Buchbergerɾվ ೖྗɿ F = (f1 , …, fs ) 13೥5݄4೔౔༵೔

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վྑ൛ Buchbergerɾվ ೖྗɿ F = (f1 , …, fs ) ॳظԽɿB := {(i, j)∣1 ≤ i < j ≤ s}ʢsyzygyجఈͷީิʣ G := F t := s 13೥5݄4೔౔༵೔

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վྑ൛ Buchbergerɾվ ೖྗɿ F = (f1 , …, fs ) ॳظԽɿB := {(i, j)∣1 ≤ i < j ≤ s}ʢsyzygyجఈͷީิʣ G := F t := s खॱɿ B = ∅ ͱͳΔ·Ͱ࣍Λ܁Γฦ͢ɿ 13೥5݄4೔౔༵೔

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վྑ൛ Buchbergerɾվ ೖྗɿ F = (f1 , …, fs ) ॳظԽɿB := {(i, j)∣1 ≤ i < j ≤ s}ʢsyzygyجఈͷީิʣ G := F t := s खॱɿ B = ∅ ͱͳΔ·Ͱ࣍Λ܁Γฦ͢ɿ 1. (i, j) ∈ B Λ೚ҙʹબͿ 13೥5݄4೔౔༵೔

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վྑ൛ Buchbergerɾվ ೖྗɿ F = (f1 , …, fs ) ॳظԽɿB := {(i, j)∣1 ≤ i < j ≤ s}ʢsyzygyجఈͷީิʣ G := F t := s खॱɿ B = ∅ ͱͳΔ·Ͱ࣍Λ܁Γฦ͢ɿ 1. (i, j) ∈ B Λ೚ҙʹબͿ 2. LCM(LT(fi ), LT(fj ))  ≠  LT(fi ) LT(fj ) ͔ͭ Test(fi , fj , B) ِ͕ͳΒ 13೥5݄4೔౔༵೔

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վྑ൛ Buchbergerɾվ ೖྗɿ F = (f1 , …, fs ) ॳظԽɿB := {(i, j)∣1 ≤ i < j ≤ s}ʢsyzygyجఈͷީิʣ G := F t := s खॱɿ B = ∅ ͱͳΔ·Ͱ࣍Λ܁Γฦ͢ɿ 1. (i, j) ∈ B Λ೚ҙʹબͿ 2. LCM(LT(fi ), LT(fj ))  ≠  LT(fi ) LT(fj ) ͔ͭ Test(fi , fj , B) ِ͕ͳΒ 1. S := S(fi , fj )G 13೥5݄4೔౔༵೔

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վྑ൛ Buchbergerɾվ ೖྗɿ F = (f1 , …, fs ) ॳظԽɿB := {(i, j)∣1 ≤ i < j ≤ s}ʢsyzygyجఈͷީิʣ G := F t := s खॱɿ B = ∅ ͱͳΔ·Ͱ࣍Λ܁Γฦ͢ɿ 1. (i, j) ∈ B Λ೚ҙʹબͿ 2. LCM(LT(fi ), LT(fj ))  ≠  LT(fi ) LT(fj ) ͔ͭ Test(fi , fj , B) ِ͕ͳΒ 1. S := S(fi , fj )G 2. S ≠ 0 ͳΒ͹ t = t + 1, ft  = S, G = G ∪ {ft }, B = B ∪ {(i, t)∣1 ≤ i ≤ t − 1} 13೥5݄4೔౔༵೔

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վྑ൛ Buchbergerɾվ ೖྗɿ F = (f1 , …, fs ) ॳظԽɿB := {(i, j)∣1 ≤ i < j ≤ s}ʢsyzygyجఈͷީิʣ G := F t := s खॱɿ B = ∅ ͱͳΔ·Ͱ࣍Λ܁Γฦ͢ɿ 1. (i, j) ∈ B Λ೚ҙʹબͿ 2. LCM(LT(fi ), LT(fj ))  ≠  LT(fi ) LT(fj ) ͔ͭ Test(fi , fj , B) ِ͕ͳΒ 1. S := S(fi , fj )G 2. S ≠ 0 ͳΒ͹ t = t + 1, ft  = S, G = G ∪ {ft }, B = B ∪ {(i, t)∣1 ≤ i ≤ t − 1} 3. B = B \ {(i, j)} 13೥5݄4೔౔༵೔

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վྑ൛ Buchbergerɾվ ೖྗɿ F = (f1 , …, fs ) ॳظԽɿB := {(i, j)∣1 ≤ i < j ≤ s}ʢsyzygyجఈͷީิʣ G := F t := s खॱɿ B = ∅ ͱͳΔ·Ͱ࣍Λ܁Γฦ͢ɿ 1. (i, j) ∈ B Λ೚ҙʹબͿ 2. LCM(LT(fi ), LT(fj ))  ≠  LT(fi ) LT(fj ) ͔ͭ Test(fi , fj , B) ِ͕ͳΒ 1. S := S(fi , fj )G 2. S ≠ 0 ͳΒ͹ t = t + 1, ft  = S, G = G ∪ {ft }, B = B ∪ {(i, t)∣1 ≤ i ≤ t − 1} 3. B = B \ {(i, j)} ग़ྗɿG 13೥5݄4೔౔༵೔

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վྑ൛ Buchbergerɾվ ೖྗɿ F = (f1 , …, fs ) ॳظԽɿB := {(i, j)∣1 ≤ i < j ≤ s}ʢsyzygyجఈͷީิʣ G := F t := s खॱɿ B = ∅ ͱͳΔ·Ͱ࣍Λ܁Γฦ͢ɿ 1. (i, j) ∈ B Λ೚ҙʹબͿ 2. LCM(LT(fi ), LT(fj ))  ≠  LT(fi ) LT(fj ) ͔ͭ Test(fi , fj , B) ِ͕ͳΒ 1. S := S(fi , fj )G 2. S ≠ 0 ͳΒ͹ t = t + 1, ft  = S, G = G ∪ {ft }, B = B ∪ {(i, t)∣1 ≤ i ≤ t − 1} 3. B = B \ {(i, j)} ग़ྗɿG ୠ͠Test(fi , fj , B) ⇔ ∃ k ≠ i, j [(i, k), (i, j) ∉ B, LT(fk ) ∣ LCM(LT(fi ), LT(fj ))] 13೥5݄4೔౔༵೔

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࣮૷ syzygyBuchberger ideal = runST $ do bases ← newSTRef { Entry LM(f) f | f ← ideal } pairs ← newSTRef {(f, g) | (f, g) ← combinations ideal } len ← newSTRef $ length ideal whileM_ (not . null <$> readSTRef pairs) $ do Just ((f, g), rest) ← viewMin <$> readSTRef pairs bases0 ← readSTRef bases b .= rest let redundant = any (λ(Entry _ h) → h ∉ {f, g} ∧ (all (∉ rest) [(f, h), (g, h)]) ∧ LM(h) | LCM(LM(f), LM(g))) bases when (LCM(LM(f), LM(g)) ≠ LM(f) LM(g) ∧ not redundant) $ do len0 ← readSTRef len let s = S(f, g) `mod` map payload (toList bases0) when (s ≠ zero) $ do pairs += { (q, s) | Entry _ q ← qs } bases %= insert (Entry LM(s) s) len *= 2 map payload . toList <$> readSTRef bases 13೥5݄4೔౔༵೔

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࣮૷ͷղઆ 13೥5݄4೔౔༵೔

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࣮૷ͷղઆ ޮ཰ͷͨΊ ST Ϟφυʹͨ͠ 13೥5݄4೔౔༵೔

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࣮૷ͷղઆ ޮ཰ͷͨΊ ST Ϟφυʹͨ͠ {} Ͱғ·Ε͍ͯΔͷ͸༏ઌ౓Ωϡʔͷͭ΋Γ 13೥5݄4೔౔༵೔

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࣮૷ͷղઆ ޮ཰ͷͨΊ ST Ϟφυʹͨ͠ {} Ͱғ·Ε͍ͯΔͷ͸༏ઌ౓Ωϡʔͷͭ΋Γ ܦݧతʹɺׂΓࢉΛ͢Δࡍ͸ઌ಄߲Λঢॱʹฒ΂͓ͯ ͘ͱΑ͍ 13೥5݄4೔౔༵೔

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࣮૷ͷղઆ ޮ཰ͷͨΊ ST Ϟφυʹͨ͠ {} Ͱғ·Ε͍ͯΔͷ͸༏ઌ౓Ωϡʔͷͭ΋Γ ܦݧతʹɺׂΓࢉΛ͢Δࡍ͸ઌ಄߲Λঢॱʹฒ΂͓ͯ ͘ͱΑ͍ ͲΜͲΜׂΕͯճ਺͕গͳ͘ͳΔ 13೥5݄4೔౔༵೔

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࣮૷ͷղઆ ޮ཰ͷͨΊ ST Ϟφυʹͨ͠ {} Ͱғ·Ε͍ͯΔͷ͸༏ઌ౓Ωϡʔͷͭ΋Γ ܦݧతʹɺׂΓࢉΛ͢Δࡍ͸ઌ಄߲Λঢॱʹฒ΂͓ͯ ͘ͱΑ͍ ͲΜͲΜׂΕͯճ਺͕গͳ͘ͳΔ جఈީิΛઌ಄߲Λ༏ઌ౓ͱ͢ΔΩϡʔͱͯ࣋ͬ͠ ͓͍ͯͯɺׂΔͱ͖ʹ toList Ͱঢॱʹฒ΂͍ͯΔ 13೥5݄4೔౔༵೔

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Benchmark simple prime syzygy 13೥5݄4೔౔༵೔

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Benchmark lex01 grlex01 grevlex01 lex03 grlex03 grevlex03 0ms 22.5ms 45ms 67.5ms 90ms simple prime syzygy grlex02 grevlex02 0s 0.375s 0.75s 1.125s 1.5s grlex04 grevlex04 0μs 150μs 300μs 450μs 600μs 13೥5݄4೔౔༵೔

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Benchmark lex01 grlex01 grevlex01 lex03 grlex03 grevlex03 0ms 22.5ms 45ms 67.5ms 90ms simple prime syzygy grlex02 grevlex02 0s 0.375s 0.75s 1.125s 1.5s grlex04 grevlex04 0μs 150μs 300μs 450μs 600μs ଎͗ͯ͢ݟ͑ͳ͍…… 13೥5݄4೔౔༵೔

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ޓ͍ʹૉ൑ఆͱͷൺֱ lex01 grlex01 grevlex01 lex03 grlex03 grevlex03 0ms 2.5ms 5ms 7.5ms 10ms grlex02 grevlex02 0s 0.375s 0.75s 1.125s 1.5s grlex04 grevlex04 58μs 59.25μs60.5μs61.75μs 63μs prime syzygy 13೥5݄4೔౔༵೔

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ޓ͍ʹૉ൑ఆͱͷൺֱ lex01 grlex01 grevlex01 lex03 grlex03 grevlex03 0ms 2.5ms 5ms 7.5ms 10ms grlex02 grevlex02 0s 0.375s 0.75s 1.125s 1.5s grlex04 grevlex04 58μs 59.25μs60.5μs61.75μs 63μs prime syzygy ͦΕͰ΋଎͍ ʂ 13೥5݄4೔౔༵೔

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঺հ͢Δख๏ θϩ؆໿ؔ܎ syzygy ͷجఈ sugar strategy 13೥5݄4೔౔༵೔

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Buchberger ͷࣗ༝౓ 13೥5݄4೔౔༵೔

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Buchberger ͷࣗ༝౓ جఈͷީิ (i, j) Λݕ౼͍ͯ͘͠ॱ൪͸ࣗ༝ 13೥5݄4೔౔༵೔

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Buchberger ͷࣗ༝౓ جఈͷީิ (i, j) Λݕ౼͍ͯ͘͠ॱ൪͸ࣗ༝ ͜ͷॱ൪ͷબͼํΛͲ͏͢Δ͔ʁ = selection strategy 13೥5݄4೔౔༵೔

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Buchberger ͷࣗ༝౓ جఈͷީิ (i, j) Λݕ౼͍ͯ͘͠ॱ൪͸ࣗ༝ ͜ͷॱ൪ͷબͼํΛͲ͏͢Δ͔ʁ = selection strategy ώϡʔϦεςΟοΫͳઓུ͕஌ΒΕ͍ͯΔ 13೥5݄4೔౔༵೔

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selection strategy 13೥5݄4೔౔༵೔

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selection strategy normal strategy 13೥5݄4೔౔༵೔

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selection strategy normal strategy ઌ಄߲ͷ LCM ͕ৗʹ࠷খʹͳΔ΋ͷ 13೥5݄4೔౔༵೔

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selection strategy normal strategy ઌ಄߲ͷ LCM ͕ৗʹ࠷খʹͳΔ΋ͷ lex ॱংͳͲ࣍਺Λอͨͳ͍ॱংͰ͸ٯޮՌ 13೥5݄4೔౔༵೔

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selection strategy normal strategy ઌ಄߲ͷ LCM ͕ৗʹ࠷খʹͳΔ΋ͷ lex ॱংͳͲ࣍਺Λอͨͳ͍ॱংͰ͸ٯޮՌ sugar strategy 13೥5݄4೔౔༵೔

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selection strategy normal strategy ઌ಄߲ͷ LCM ͕ৗʹ࠷খʹͳΔ΋ͷ lex ॱংͳͲ࣍਺Λอͨͳ͍ॱংͰ͸ٯޮՌ sugar strategy Ծ૝తͳ੪࣍Խ࣍਺͕࠷খͷ෺ΛબͿ 13೥5݄4೔౔༵೔

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selection strategy normal strategy ઌ಄߲ͷ LCM ͕ৗʹ࠷খʹͳΔ΋ͷ lex ॱংͳͲ࣍਺Λอͨͳ͍ॱংͰ͸ٯޮՌ sugar strategy Ծ૝తͳ੪࣍Խ࣍਺͕࠷খͷ෺ΛબͿ ͍ͣΕ΋ಉ཰ҰҐ͕ग़ͨΒૣ͘ग़ͯདྷͨํΛબͿ 13೥5݄4೔౔༵೔

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selection strategy normal strategy ઌ಄߲ͷ LCM ͕ৗʹ࠷খʹͳΔ΋ͷ lex ॱংͳͲ࣍਺Λอͨͳ͍ॱংͰ͸ٯޮՌ sugar strategy Ծ૝తͳ੪࣍Խ࣍਺͕࠷খͷ෺ΛબͿ ͍ͣΕ΋ಉ཰ҰҐ͕ग़ͨΒૣ͘ग़ͯདྷͨํΛબͿ ͜ͷ৚݅Λൈ͔͢ͱڪ͘͠஗͘ͳΔ…… 13೥5݄4೔౔༵೔

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੪࣍Խ࣍਺ 13೥5݄4೔౔༵೔

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੪࣍Խ࣍਺ ੪࣍ࣜʢ֤୯߲ࣜͷશ࣍਺͕౳͍͠ଟ߲ࣜʣΛੜ੒ݩͱ͢ΔΠ σΞϧͷܭࢉ͸ܦݧతʹ଎͍ 13೥5݄4೔౔༵೔

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੪࣍Խ࣍਺ ੪࣍ࣜʢ֤୯߲ࣜͷશ࣍਺͕౳͍͠ଟ߲ࣜʣΛੜ੒ݩͱ͢ΔΠ σΞϧͷܭࢉ͸ܦݧతʹ଎͍ ༨෼ͳม਺Ͱੜ੒ݩΛ੪࣍Խ͔ͯ͠Βܭࢉ͢Δͱ͍͏ώϡʔ ϦεςΟΫε͕஌ΒΕ͍ͯΔ 13೥5݄4೔౔༵೔

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੪࣍Խ࣍਺ ੪࣍ࣜʢ֤୯߲ࣜͷશ࣍਺͕౳͍͠ଟ߲ࣜʣΛੜ੒ݩͱ͢ΔΠ σΞϧͷܭࢉ͸ܦݧతʹ଎͍ ༨෼ͳม਺Ͱੜ੒ݩΛ੪࣍Խ͔ͯ͠Βܭࢉ͢Δͱ͍͏ώϡʔ ϦεςΟΫε͕஌ΒΕ͍ͯΔ ੪࣍Խͷྫɿxy + 2x + y3 + 3  ⇒  xyz + 2xz2 + y3 + 3z3 13೥5݄4೔౔༵೔

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੪࣍Խ࣍਺ ੪࣍ࣜʢ֤୯߲ࣜͷશ࣍਺͕౳͍͠ଟ߲ࣜʣΛੜ੒ݩͱ͢ΔΠ σΞϧͷܭࢉ͸ܦݧతʹ଎͍ ༨෼ͳม਺Ͱੜ੒ݩΛ੪࣍Խ͔ͯ͠Βܭࢉ͢Δͱ͍͏ώϡʔ ϦεςΟΫε͕஌ΒΕ͍ͯΔ ੪࣍Խͷྫɿxy + 2x + y3 + 3  ⇒  xyz + 2xz2 + y3 + 3z3 ໰୊఺ɿಘΒΕͨجఈ͔Β༨෼ͳม਺Λফڈͯ͠΋ Gröbner جఈʹͳΔͱ͸ݶΒͳ͍ 13೥5݄4೔౔༵೔

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੪࣍Խ࣍਺ ੪࣍ࣜʢ֤୯߲ࣜͷશ࣍਺͕౳͍͠ଟ߲ࣜʣΛੜ੒ݩͱ͢ΔΠ σΞϧͷܭࢉ͸ܦݧతʹ଎͍ ༨෼ͳม਺Ͱੜ੒ݩΛ੪࣍Խ͔ͯ͠Βܭࢉ͢Δͱ͍͏ώϡʔ ϦεςΟΫε͕஌ΒΕ͍ͯΔ ੪࣍Խͷྫɿxy + 2x + y3 + 3  ⇒  xyz + 2xz2 + y3 + 3z3 ໰୊఺ɿಘΒΕͨجఈ͔Β༨෼ͳม਺Λফڈͯ͠΋ Gröbner جఈʹͳΔͱ͸ݶΒͳ͍ ੪࣍Խͯ͠ಘΒΕͨݩʹରͯ͠΋͏Ұ౓ Buchberger Λ࢖͏͜ͱʹͳΔʢͦΕͰ΋ଟগ͸଎͘ͳΔ৔߹͕ ͋ΔΒ͍͠ʣ 13೥5݄4೔౔༵೔

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sugar strategy 13೥5݄4೔౔༵೔

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sugar strategy ී௨ʹ Buchberger Λܭࢉ͢Δ͕ɺ(i, j) Λબ ୒͢Δͱ͖͚ͩԾ૝తͳ੪࣍ԽΛߦ͍ɺͦͷ ࣍਺࠷খͷ΋ͷΛબͿ 13೥5݄4೔౔༵೔

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sugar strategy ී௨ʹ Buchberger Λܭࢉ͢Δ͕ɺ(i, j) Λબ ୒͢Δͱ͖͚ͩԾ૝తͳ੪࣍ԽΛߦ͍ɺͦͷ ࣍਺࠷খͷ΋ͷΛબͿ ͜ͷԾ૝తͳ࣍਺ͷ͜ͱΛ sugar ͱ͍͏ 13೥5݄4೔౔༵೔

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sugar ଟ߲ࣜ (fi , fj ) ͷ sugar Λ࣍ͷΑ͏ʹఆΊΔɻ 13೥5݄4೔౔༵೔

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sugar ଟ߲ࣜ (fi , fj ) ͷ sugar Λ࣍ͷΑ͏ʹఆΊΔɻ Sfᵢ  = deg(fi ), Sfⱼ  = deg(fj ) 13೥5݄4೔౔༵೔

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sugar ଟ߲ࣜ (fi , fj ) ͷ sugar Λ࣍ͷΑ͏ʹఆΊΔɻ Sfᵢ  = deg(fi ), Sfⱼ  = deg(fj ) Sf ɿgiven Ͱ t ͕୯߲ࣜͳΒ Stf  = deg(t) + Sf 13೥5݄4೔౔༵೔

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sugar ଟ߲ࣜ (fi , fj ) ͷ sugar Λ࣍ͷΑ͏ʹఆΊΔɻ Sfᵢ  = deg(fi ), Sfⱼ  = deg(fj ) Sf ɿgiven Ͱ t ͕୯߲ࣜͳΒ Stf  = deg(t) + Sf f = g + h ͷ࣌ɺSf  = max(Sg , Sf ) 13೥5݄4೔౔༵೔

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sugar ଟ߲ࣜ (fi , fj ) ͷ sugar Λ࣍ͷΑ͏ʹఆΊΔɻ Sfᵢ  = deg(fi ), Sfⱼ  = deg(fj ) Sf ɿgiven Ͱ t ͕୯߲ࣜͳΒ Stf  = deg(t) + Sf f = g + h ͷ࣌ɺSf  = max(Sg , Sf ) (fi , fj ) ͷ sugar ͸ɺ্ʹैͬͯܭࢉͨ͠ S-ଟ ߲ࣜͷ sugar ͱ͢Δɻ 13೥5݄4೔౔༵೔

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࣮ࡍͷ࣮૷ 13೥5݄4೔౔༵೔

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࣮ࡍͷ࣮૷ ͓͋ͭΒ͑޲͖ʹ༏ઌ౓ΩϡʔΛ࢖͍ͬͯͨ 13೥5݄4೔౔༵೔

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࣮ࡍͷ࣮૷ ͓͋ͭΒ͑޲͖ʹ༏ઌ౓ΩϡʔΛ࢖͍ͬͯͨ “selection strategy = ༏ઌ౓ͷܭࢉํ๏” ͱͯ͠ந৅Խग़དྷΔʂ 13೥5݄4೔౔༵೔

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࣮ࡍͷ࣮૷ ͓͋ͭΒ͑޲͖ʹ༏ઌ౓ΩϡʔΛ࢖͍ͬͯͨ “selection strategy = ༏ઌ౓ͷܭࢉํ๏” ͱͯ͠ந৅Խग़དྷΔʂ double sugar ͱ͍͏ख๏͸͜Ε͚ͩͰ͸ແཧ ʢͰ΋ͦ͜·ͰޮՌ͕ͳ͍Β͍͠ͷͰແࢹʣ 13೥5݄4೔౔༵೔

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࣮ࡍͷ࣮૷ ͓͋ͭΒ͑޲͖ʹ༏ઌ౓ΩϡʔΛ࢖͍ͬͯͨ “selection strategy = ༏ઌ౓ͷܭࢉํ๏” ͱͯ͠ந৅Խग़དྷΔʂ double sugar ͱ͍͏ख๏͸͜Ε͚ͩͰ͸ແཧ ʢͰ΋ͦ͜·ͰޮՌ͕ͳ͍Β͍͠ͷͰແࢹʣ (i, j) Λอଘ͢Δ࣌ʹॏΈ΋อଘ͓ͯ͘͠ 13೥5݄4೔౔༵೔

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࣮૷ syzygyBuchbergerWithStrategy strategy ideal = runST $ do let gens = zip [1..] ideal bases ← newSTRef { Entry LM(f) f | (_, f) ← gens } pairs ← newSTRef { Entry (calcWeight strategy f g, j) (f, g) | ((i, f), (j, g)) ← combinations ideal } len ← newSTRef $ length ideal whileM_ (not . null <$> readSTRef pairs) $ do Just (Entry _ (f, g), rest) ← viewMin <$> readSTRef pairs -- தུ when (s ≠ zero) $ do pairs += { Entry (calcWeight strategy q s, j) (q, s) | Entry _ q ← qs | j ← [len0 + 1 ..] } bases %= insert (Entry LM(s) s) len *= 2 map payload . toList <$> readSTRef bases ॏΈΛ௥Ճ͢Δ͚ͩͰ࣮૷ग़དྷͯ͠·ͬͨ…… 13೥5݄4೔౔༵೔

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Benchmark syzygy sugar 13೥5݄4೔౔༵೔

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Benchmark lex01 grlex01 grevlex01 lex03 grlex03 grevlex03 0ms 0.15ms 0.3ms 0.45ms 0.6ms grlex02 grevlex02 16ms 17ms 18ms 19ms 20ms grlex04 grevlex04 58μs 58.1μs 58.2μs58.3μs58.4μs syzygy sugar 13೥5݄4೔౔༵೔

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Benchmarkʢ࿦จ൛ʣ syzygy sugar I1  = ⟨35y4 − 10xy2 − 210y2z + 3x2 + 30xz − 105z2 + 140yt − 21u,  5xy3 − 140y3z − 3x2y + 45xyz − 420yz2 + 210y2t − 25xt + 70zt + 126yu⟩ I2  = ⟨x + y + z + w, xy + yz + zw + wx, xyz + yzw + zwx + wxy, xyzw − 1⟩ɹ I3  = ⟨x31 − x6 − x − y, x8 − z, x10 − t⟩ 13೥5݄4೔౔༵೔

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Benchmarkʢ࿦จ൛ʣ grevlex01 lex02 grevlex02 0ms 10ms 20ms 30ms 40ms grevlex03 0ms 700ms1400ms2100ms 2800ms syzygy sugar lex01 0ms 0.5ms 1ms 1.5ms 2ms I1  = ⟨35y4 − 10xy2 − 210y2z + 3x2 + 30xz − 105z2 + 140yt − 21u,  5xy3 − 140y3z − 3x2y + 45xyz − 420yz2 + 210y2t − 25xt + 70zt + 126yu⟩ I2  = ⟨x + y + z + w, xy + yz + zw + wx, xyz + yzw + zwx + wxy, xyzw − 1⟩ɹ I3  = ⟨x31 − x6 − x − y, x8 − z, x10 − t⟩ 13೥5݄4೔౔༵೔

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ߟ࡯ 13೥5݄4೔౔༵೔

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ߟ࡯ ࿦จʹग़ͯ͘Δྫʹ͍ͭͯ͸ sugar ͕଎͍ 13೥5݄4೔౔༵೔

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ߟ࡯ ࿦จʹग़ͯ͘Δྫʹ͍ͭͯ͸ sugar ͕଎͍ ͦΕͰ΋ڻ͘΄Ͳ଎͍Α͏ʹ͸ݟ͑ͳ͍ 13೥5݄4೔౔༵೔

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ߟ࡯ ࿦จʹग़ͯ͘Δྫʹ͍ͭͯ͸ sugar ͕଎͍ ͦΕͰ΋ڻ͘΄Ͳ଎͍Α͏ʹ͸ݟ͑ͳ͍ ͪ͜ΒͰબΜͩσʔληοτʹؔͯ͠͸ sugar ͷ΄͏͕஗͍৔߹΋͋Δʁ 13೥5݄4೔౔༵೔

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ߟ࡯ ࿦จʹग़ͯ͘Δྫʹ͍ͭͯ͸ sugar ͕଎͍ ͦΕͰ΋ڻ͘΄Ͳ଎͍Α͏ʹ͸ݟ͑ͳ͍ ͪ͜ΒͰબΜͩσʔληοτʹؔͯ͠͸ sugar ͷ΄͏͕஗͍৔߹΋͋Δʁ ௥੻ௐࠪ༧ఆ 13೥5݄4೔౔༵೔

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͜͜·ͰͰ࣭໰ʁ 13೥5݄4೔౔༵೔

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Table of Contents Gröbner جఈͷ؆୯ͳಋೖ Gröbner جఈͷܭࢉ๏ Buchberger ΞϧΰϦζϜ ࠷దԽख๏ɿsyzygy, sugar strategy Haskell ͷܕϨϕϧϓϩάϥϛϯάͷݱࡏ 13೥5݄4೔౔༵೔

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Haskell ͷ ܕϨϕϧϓϩάϥϛϯάͷݱࡏ 13೥5݄4೔౔༵೔

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Haskell ͷ ܕϨϕϧϓϩάϥϛϯάͷݱࡏ computational-algebra Ͱ͸ܕػೳΛ;ΜͩΜʹ ར༻͍ͯ͠Δ 13೥5݄4೔౔༵೔

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Haskell ͷ ܕϨϕϧϓϩάϥϛϯάͷݱࡏ computational-algebra Ͱ͸ܕػೳΛ;ΜͩΜʹ ར༻͍ͯ͠Δ ม਺ͷ਺ɺମͷछྨɺ୯߲ࣜॱংɺબ୒ઓུ… 13೥5݄4೔౔༵೔

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Haskell ͷ ܕϨϕϧϓϩάϥϛϯάͷݱࡏ computational-algebra Ͱ͸ܕػೳΛ;ΜͩΜʹ ར༻͍ͯ͠Δ ม਺ͷ਺ɺମͷछྨɺ୯߲ࣜॱংɺબ୒ઓུ… શͯܕʹύϥϝτϥΠζ 13೥5݄4೔౔༵೔

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Haskell ͷ ܕϨϕϧϓϩάϥϛϯάͷݱࡏ computational-algebra Ͱ͸ܕػೳΛ;ΜͩΜʹ ར༻͍ͯ͠Δ ม਺ͷ਺ɺମͷछྨɺ୯߲ࣜॱংɺબ୒ઓུ… શͯܕʹύϥϝτϥΠζ ґଘܕͱূ໌ͷΦϯύϨʔυ 13೥5݄4೔౔༵೔

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ґଘܕͱূ໌ͷ ΦϯύϨʔυ 13೥5݄4೔౔༵೔

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ґଘܕͱূ໌ͷ ΦϯύϨʔυ DataKinds ΍ GADTs ͱ͍֦ͬͨுػೳΛଟ༻ 13೥5݄4೔౔༵೔

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ґଘܕͱূ໌ͷ ΦϯύϨʔυ DataKinds ΍ GADTs ͱ͍֦ͬͨுػೳΛଟ༻ ୯߲ࣜ͸ࢦ਺ϕΫτϧͰදݱ͍ͯ͠ΔͷͰɺม਺ͷ਺ͷ۠ ผΛ͔ͬ͠Γ͚͍ͭͨ 13೥5݄4೔౔༵೔

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ґଘܕͱূ໌ͷ ΦϯύϨʔυ DataKinds ΍ GADTs ͱ͍֦ͬͨுػೳΛଟ༻ ୯߲ࣜ͸ࢦ਺ϕΫτϧͰදݱ͍ͯ͠ΔͷͰɺม਺ͷ਺ͷ۠ ผΛ͔ͬ͠Γ͚͍ͭͨ ௕͖ͭ͞ϕΫτϧʂ 13೥5݄4೔౔༵೔

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ґଘܕͱূ໌ͷ ΦϯύϨʔυ DataKinds ΍ GADTs ͱ͍֦ͬͨுػೳΛଟ༻ ୯߲ࣜ͸ࢦ਺ϕΫτϧͰදݱ͍ͯ͠ΔͷͰɺม਺ͷ਺ͷ۠ ผΛ͔ͬ͠Γ͚͍ͭͨ ௕͖ͭ͞ϕΫτϧʂ ڥք৚݅΋ূ໌͠Α͏ʂ 13೥5݄4೔౔༵೔

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ґଘܕͱূ໌ͷ ΦϯύϨʔυ DataKinds ΍ GADTs ͱ͍֦ͬͨுػೳΛଟ༻ ୯߲ࣜ͸ࢦ਺ϕΫτϧͰදݱ͍ͯ͠ΔͷͰɺม਺ͷ਺ͷ۠ ผΛ͔ͬ͠Γ͚͍ͭͨ ௕͖ͭ͞ϕΫτϧʂ ڥք৚݅΋ূ໌͠Α͏ʂ ͜Ε͚ͩͰ΋͔ͳΓେมͳͷͰɺʮΞϧΰϦζϜͷ ଥ౰ੑʯͱ͔͸͍ࣔͯ͠ͳ͍Ͱ͢ʢࢮ͵ʣ 13೥5݄4೔౔༵೔

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ৄ͘͠͸ίʔυࢀর -- | N-ary Monomial. IntMap contains degrees for each xi. type Monomial (n :: Nat) = Vector Int n -- | Monomial with monomial order. newtype OrderedMonomial (ord :: ⋆) n = OrderedMonomial { getMonomial :: Monomial n } deriving instance (Eq (Monomial n)) ⇒  Eq (OrderedMonomial ordering n) type MonomialOrder = ∀n. Monomial n → Monomial n → Ordering -- | Class to lookup ordering from its (type-level) name. class IsOrder (ordering :: ⋆) where cmpMonomial :: Proxy ordering → MonomialOrder -- | Class for Monomial orders. class IsOrder name ⇒ IsMonomialOrder name where 13೥5݄4೔౔༵೔

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ৄ͘͠͸ίʔυࢀর data Lex = Lex data Revlex = Revlex data Grevlex = Grevlex lex :: MonomialOrder lex Nil Nil = EQ lex (x :- xs) (y :- ys) = x `compare` y <> xs `lex` ys lex _ _ = error "cannot happen" revlex :: Monomial n → Monomial n → Ordering revlex (x :- xs) (y :- ys) = xs `revlex` ys <> y `compare` x revlex Nil Nil = EQ revlex _ _ = error "cannot happen!" graded :: (Monomial n → Monomial n → Ordering) → (Monomial n → Monomial n → Ordering) graded cmp xs ys = comparing totalDegree xs ys <> cmp xs ys instance IsOrder Grevlex where cmpMonomial _ = graded revlex instance IsOrder Revlex where cmpMonomial _ = revlex 13೥5݄4೔౔༵೔

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cannot happen! 13೥5݄4೔౔༵೔

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cannot happen! ܕϨϕϧͰېࢭ͞ΕΔύλʔϯ 13೥5݄4೔౔༵೔

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cannot happen! ܕϨϕϧͰېࢭ͞ΕΔύλʔϯ ͔͠͠ɺ -fwarn-incomplete-pattern Λࢦఆ ͢Δͱʮ͜ͷύλʔϯൈ͚ͯΔΑʂʯ 13೥5݄4೔౔༵೔

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cannot happen! ܕϨϕϧͰېࢭ͞ΕΔύλʔϯ ͔͠͠ɺ -fwarn-incomplete-pattern Λࢦఆ ͢Δͱʮ͜ͷύλʔϯൈ͚ͯΔΑʂʯ ͦͷύλʔϯΛॻ͘ͱܕΤϥʔͰ஄͔ΕΔ 13೥5݄4೔౔༵೔

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cannot happen! ܕϨϕϧͰېࢭ͞ΕΔύλʔϯ ͔͠͠ɺ -fwarn-incomplete-pattern Λࢦఆ ͢Δͱʮ͜ͷύλʔϯൈ͚ͯΔΑʂʯ ͦͷύλʔϯΛॻ͘ͱܕΤϥʔͰ஄͔ΕΔ ͔ͩΒ wildcard pattern 13೥5݄4೔౔༵೔

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ΠσΞϧ hack data Ideal r = ∀n. Ideal (Vector r n) instance Eq r ⇒ Eq (Ideal r) where (≡) = (≡) `on` generators generators :: Ideal r → [r] generators (Ideal is) = toList is addToIdeal :: r → Ideal r → Ideal r addToIdeal i (Ideal is) = Ideal (i :- is) toIdeal :: NoetherianRing r ⇒ [r] → Ideal r toIdeal = foldr addToIdeal (Ideal Nil) 13೥5݄4೔౔༵೔

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ΠσΞϧ hack data Ideal r = ∀n. Ideal (Vector r n) instance Eq r ⇒ Eq (Ideal r) where (≡) = (≡) `on` generators generators :: Ideal r → [r] generators (Ideal is) = toList is addToIdeal :: r → Ideal r → Ideal r addToIdeal i (Ideal is) = Ideal (i :- is) toIdeal :: NoetherianRing r ⇒ [r] → Ideal r toIdeal = foldr addToIdeal (Ideal Nil) ΠσΞϧ͸Ϧετ……Ͱ͸ͳ͍ʂ 13೥5݄4೔౔༵೔

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ΠσΞϧ hack data Ideal r = ∀n. Ideal (Vector r n) instance Eq r ⇒ Eq (Ideal r) where (≡) = (≡) `on` generators generators :: Ideal r → [r] generators (Ideal is) = toList is addToIdeal :: r → Ideal r → Ideal r addToIdeal i (Ideal is) = Ideal (i :- is) toIdeal :: NoetherianRing r ⇒ [r] → Ideal r toIdeal = foldr addToIdeal (Ideal Nil) ΠσΞϧ͸Ϧετ……Ͱ͸ͳ͍ʂ ඇৗʹਂԕͳΔཧ༝ʹΑΓଘࡏྔԽ͞Εͨ sized-vector 13೥5݄4೔౔༵೔

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ΠσΞϧ hack data Ideal r = ∀n. Ideal (Vector r n) instance Eq r ⇒ Eq (Ideal r) where (≡) = (≡) `on` generators generators :: Ideal r → [r] generators (Ideal is) = toList is addToIdeal :: r → Ideal r → Ideal r addToIdeal i (Ideal is) = Ideal (i :- is) toIdeal :: NoetherianRing r ⇒ [r] → Ideal r toIdeal = foldr addToIdeal (Ideal Nil) ΠσΞϧ͸Ϧετ……Ͱ͸ͳ͍ʂ ඇৗʹਂԕͳΔཧ༝ʹΑΓଘࡏྔԽ͞Εͨ sized-vector ফڈΠσΞϧͷܭࢉͷ࣌ʹ҉໧ཪʹܕΛ߹ΘͤΔҝ 13೥5݄4೔౔༵೔

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⁋᧒Λ߹Θͤʹ༻ҙ͞Εͨ data Monomorphic k = ∀a. Monomorphic (k a) -- | A types which have the monomorphic representation. class Monomorphicable k where -- | Monomorphic representation type MonomorphicRep k :: ⋆ -- | Promote the monomorphic value to the polymophic one. promote :: MonomorphicRep k → Monomorphic k -- | Demote the polymorphic value to the monomorphic representation. demote :: Monomorphic k → MonomorphicRep k -- | Convinience function to demote polymorphic types -- into monomorphic one directly. demote' :: Monomorphicable k ⇒ k a → MonomorphicRep k demote' = demote ∘ Monomorphic -- | Demote polymorphic nested types directly into -- monomorphic representation. demoteComposed :: Monomorphicable (f :∘: g) ⇒ f (g a) → MonomorphicRep (f :∘: g) demoteComposed = demote ∘ Monomorphic ∘ Comp 13೥5݄4೔౔༵೔

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੍໿෇͚ͯΔײ͡ thEliminationIdeal :: ( IsMonomialOrder ord, Field k, IsPolynomial k m , IsPolynomial k (m :-: n) , (n ≼ m) ~ True) ⇒ SNat n → Ideal (OrderedPolynomial k ord m) → Ideal (OrderedPolynomial k ord (m ⊖ n)) thEliminationIdeal n = case singInstance n of SingInstance → mapIdeal (changeOrderProxy Proxy) ∘ thEliminationIdealWith (weightedEliminationOrder n) n 13೥5݄4೔౔༵೔

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੍໿෇͚ͯΔײ͡ thEliminationIdeal :: ( IsMonomialOrder ord, Field k, IsPolynomial k m , IsPolynomial k (m :-: n) , (n ≼ m) ~ True) ⇒ SNat n → Ideal (OrderedPolynomial k ord m) → Ideal (OrderedPolynomial k ord (m ⊖ n)) thEliminationIdeal n = case singInstance n of SingInstance → mapIdeal (changeOrderProxy Proxy) ∘ thEliminationIdealWith (weightedEliminationOrder n) n Gröbner جఈΛ࢖͏ͱɺ࿈ཱํఔ͔ࣜΒจࣈΛফ ڈग़དྷΔʢ͍͢͝ʂʣ 13೥5݄4೔౔༵೔

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੍໿෇͚ͯΔײ͡ thEliminationIdeal :: ( IsMonomialOrder ord, Field k, IsPolynomial k m , IsPolynomial k (m :-: n) , (n ≼ m) ~ True) ⇒ SNat n → Ideal (OrderedPolynomial k ord m) → Ideal (OrderedPolynomial k ord (m ⊖ n)) thEliminationIdeal n = case singInstance n of SingInstance → mapIdeal (changeOrderProxy Proxy) ∘ thEliminationIdealWith (weightedEliminationOrder n) n Gröbner جఈΛ࢖͏ͱɺ࿈ཱํఔ͔ࣜΒจࣈΛফ ڈग़དྷΔʢ͍͢͝ʂʣ n จࣈ໨·ͰΛফڈ͍ͨ͠㱺 ม਺͸࠷௿Ͱ΋ n ݸ 13೥5݄4೔౔༵೔

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ূ໌ͯ͠Δײ͡ intersection :: ∀ r k n ord. ( IsMonomialOrder ord, Field r, IsPolynomial r k, IsPolynomial r n , IsPolynomial r (k ⊕ n) ) ⇒ Vector (Ideal (OrderedPolynomial r ord n)) k → Ideal (OrderedPolynomial r ord n) intersection Nil = Ideal $ singletonV one intersection idsv@(_ :- _) = let sk = sLengthV idsv sn = sing :: SNat n ts = genVars (sk %+ sn) tis = zipWith (λ ideal t → mapIdeal ((t *) . shiftR sk) ideal) (toList idsv) ts j = foldr appendIdeal (principalIdeal (one - foldr (+) zero ts)) tis in case plusMinusEqR sn sk of Eql → case propToBoolLeq (plusLeqL sk sn) of LeqTrueInstance → thEliminationIdeal sk j 13೥5݄4೔౔༵೔

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ࠓؾ෇͍͚ͨͲ શ෦આ໌͢Δͷແཧͩ 13೥5݄4೔౔༵೔

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΋͏ͩΊͩ 13೥5݄4೔౔༵೔

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ৄ͘͠͸௚઀ั·͑ͯ 㘤͍͍ͯͩ͘͞ 13೥5݄4೔౔༵೔

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࠷ޙʹ͍͍͍ͨ͜ͱ 13೥5݄4೔౔༵೔

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࠷ޙʹ͍͍͍ͨ͜ͱ ؼೲ๏͸ίετ͕͔͔Δ 13೥5݄4೔౔༵೔

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࠷ޙʹ͍͍͍ͨ͜ͱ ؼೲ๏͸ίετ͕͔͔Δ ূ໌͸࣮ߦ࣌ʹ஗Ԇ͞ΕΔ͔Βλμ͡Όͳ͍ 13೥5݄4೔౔༵೔

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࠷ޙʹ͍͍͍ͨ͜ͱ ؼೲ๏͸ίετ͕͔͔Δ ূ໌͸࣮ߦ࣌ʹ஗Ԇ͞ΕΔ͔Βλμ͡Όͳ͍ γϯάϧτϯ΋ίετ͕͔͔Δ 13೥5݄4೔౔༵೔

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࠷ޙʹ͍͍͍ͨ͜ͱ ؼೲ๏͸ίετ͕͔͔Δ ূ໌͸࣮ߦ࣌ʹ஗Ԇ͞ΕΔ͔Βλμ͡Όͳ͍ γϯάϧτϯ΋ίετ͕͔͔Δ Haskell ͰͷґଘܕͰ࣮༻తͳΞϓϦॻ͘ͷ͸Ή͍ͣ 13೥5݄4೔౔༵೔

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࠷ޙʹ͍͍͍ͨ͜ͱ ؼೲ๏͸ίετ͕͔͔Δ ূ໌͸࣮ߦ࣌ʹ஗Ԇ͞ΕΔ͔Βλμ͡Όͳ͍ γϯάϧτϯ΋ίετ͕͔͔Δ Haskell ͰͷґଘܕͰ࣮༻తͳΞϓϦॻ͘ͷ͸Ή͍ͣ HP ඪ४ఴ෇ͷ GHC 7.4.* ͸Ϋι 13೥5݄4೔౔༵೔

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࠷ޙʹ͍͍͍ͨ͜ͱ ؼೲ๏͸ίετ͕͔͔Δ ূ໌͸࣮ߦ࣌ʹ஗Ԇ͞ΕΔ͔Βλμ͡Όͳ͍ γϯάϧτϯ΋ίετ͕͔͔Δ Haskell ͰͷґଘܕͰ࣮༻తͳΞϓϦॻ͘ͷ͸Ή͍ͣ HP ඪ४ఴ෇ͷ GHC 7.4.* ͸Ϋι ίϯύΠϧ͢Δ౓ʹ rm *.hi *.o ͠ͳ͍ͱܕػೳ͕࢖ ͑ͳ͍ 13೥5݄4೔౔༵೔

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࠷ޙʹ͍͍͍ͨ͜ͱ ؼೲ๏͸ίετ͕͔͔Δ ূ໌͸࣮ߦ࣌ʹ஗Ԇ͞ΕΔ͔Βλμ͡Όͳ͍ γϯάϧτϯ΋ίετ͕͔͔Δ Haskell ͰͷґଘܕͰ࣮༻తͳΞϓϦॻ͘ͷ͸Ή͍ͣ HP ඪ४ఴ෇ͷ GHC 7.4.* ͸Ϋι ίϯύΠϧ͢Δ౓ʹ rm *.hi *.o ͠ͳ͍ͱܕػೳ͕࢖ ͑ͳ͍ छଟ૬ͳ֎෦ϥΠϒϥϦͱϦϯΫग़དྷͳ͍ 13೥5݄4೔౔༵೔

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࠷ޙʹ͍͍͍ͨ͜ͱ ؼೲ๏͸ίετ͕͔͔Δ ূ໌͸࣮ߦ࣌ʹ஗Ԇ͞ΕΔ͔Βλμ͡Όͳ͍ γϯάϧτϯ΋ίετ͕͔͔Δ Haskell ͰͷґଘܕͰ࣮༻తͳΞϓϦॻ͘ͷ͸Ή͍ͣ HP ඪ४ఴ෇ͷ GHC 7.4.* ͸Ϋι ίϯύΠϧ͢Δ౓ʹ rm *.hi *.o ͠ͳ͍ͱܕػೳ͕࢖ ͑ͳ͍ छଟ૬ͳ֎෦ϥΠϒϥϦͱϦϯΫग़དྷͳ͍ Ҋ֎ܕͷ⁋᧒͸͋͏ 13೥5݄4೔౔༵೔

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·ͱΊ 13೥5݄4೔౔༵೔

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·ͱΊ GröbnerجఈΛ࢖͏ͱ୅਺ͷܭࢉ͕ग़དྷΔΑ͏ʹ 13೥5݄4೔౔༵೔

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·ͱΊ GröbnerجఈΛ࢖͏ͱ୅਺ͷܭࢉ͕ग़དྷΔΑ͏ʹ ΠσΞϧ঎ͱ͔ڞ௨෦෼ͱ͔ 13೥5݄4೔౔༵೔

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·ͱΊ GröbnerجఈΛ࢖͏ͱ୅਺ͷܭࢉ͕ग़དྷΔΑ͏ʹ ΠσΞϧ঎ͱ͔ڞ௨෦෼ͱ͔ จࣈফڈɺ࿈ཱํఔࣜͷղ 13೥5݄4೔౔༵೔

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·ͱΊ GröbnerجఈΛ࢖͏ͱ୅਺ͷܭࢉ͕ग़དྷΔΑ͏ʹ ΠσΞϧ঎ͱ͔ڞ௨෦෼ͱ͔ จࣈফڈɺ࿈ཱํఔࣜͷղ ߴ଎Խख๏͕ز͔ͭ͋ͬͯɺ੾᛭ୖຏ 13೥5݄4೔౔༵೔

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·ͱΊ GröbnerجఈΛ࢖͏ͱ୅਺ͷܭࢉ͕ग़དྷΔΑ͏ʹ ΠσΞϧ঎ͱ͔ڞ௨෦෼ͱ͔ จࣈফڈɺ࿈ཱํఔࣜͷղ ߴ଎Խख๏͕ز͔ͭ͋ͬͯɺ੾᛭ୖຏ Haskell ܕγεςϜͷࠓޙʹظ଴ 13೥5݄4೔౔༵೔

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ࢀߟจݙ D. Cox, J. Little and D. O’Shea. Ideals, Varieties and Algorithms. UTM. Springer. A. Giovini, T. Moran, G. Niesi, L. Robbiano and C. Traverse. "One sugar cube, please" OR Selection strategies in the Buchberger algorithm. ᑸݩ. άϨϒφʔجఈ͸໘ന͍ʂ. ߨٛ࿥. 2012೥౓ૣҴాେֶجװཧ޻ֶ෦਺ֶՊ୅਺ֶC1 ߨٛࢿྉ 13೥5݄4೔౔༵೔

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Any Questions? 13೥5݄4೔౔༵೔

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ޚਗ਼ௌ ͋Γ͕ͱ͏͍͟͝·ͨ͠ 13೥5݄4೔౔༵೔