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Reduceを使った定理証明 〜 グレブナー基底を添えて 〜
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Qiushi Pan
December 10, 2018
Science
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Reduceを使った定理証明 〜 グレブナー基底を添えて 〜
主専攻実習,定理証明グループ最終発表スライド.Reduceを使った定理証明の解説を,グレブナー基底から,具体的な定理で証明するまで.
Qiushi Pan
December 10, 2018
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Transcript
ఆཧূ໌άϧʔϓ 5IFPSFN1SPWJOH1SBDUJDF !RRIBOO
શମઆ໌
ϓϩάϥϜͰఆཧূ໌ʁ w ͦΕͬͯͲ͏Δͷʁ͓͍͍͠ͷʁ w ԾఆΛஔ͍ͨ࣌ʹɼ͕݁ਖ਼͍͜͠ͱΛࣔͤΕྑ͍ɽ
४උάϨϒφʔجఈ w ํఔࣜ !" , … , !% → άϨϒφʔجఈ
'" , … , '( w ํఔࣜΛղ͖͍͢ܗʹมͨ͠ͷ w ଟมଟ߲ࣜͷআࢉ ֤εςοϓΛ؆ͱΈͳ͢ ʢ.؆ SFEVDJUPO PG.VMUJQMFTʣ
४උΠσΞϧ w ՄRͷΠσΞϧI Rؚ͕ΉIͰҎԼΛຬͨ͢ͷ i) %, ' ∈ ) ⟹
% ± ' ∈ ) ii) % ∈ ), , ∈ - ⟹ %, ∈ ) w ྫɿ R = ℤ, I={3} 3) 34 ± 35 = 3 4 ± 5 33) 34 5 = 3(45)
w ΠσΞϧ ͷഒ Λ 3 ͱॻ͘ɽ 9 ∈ (3) 8
∉ 3 9 ÷ 3 = 3 ⋯ 0 ͷഒ 8 ÷ 3 = 2 ⋯ 2 ͷഒͰͳ͍ w Ծఆ-. , … , -1 ݁ 2 2 34,…,35 0 ͳΒ 2 ∈ -. , … , -1 Γཱͭอূ͕ͳ͍㱺άϨϒφʔجఈ ℎ. , … , ℎ7 ʹ࡞ Γม͑Δ 2 84,…,89 0 ͳΒ 2 ∈ -. , … , -1 ఆཧΛࣔ͢खॱ
ఆཧূ໌ ʰϞϊάϥϑ زԿֶʱ ݈ଠɼਗ਼ٶढ़༤ɽ Qͷఆཧ
ʮఆཧ ਖ਼ํܗ"#$%ͷล$%্ͷΛ&ͱ͢Δɽ&͔Β "$ʹԼΖͨ͠ਨઢͷΛ1ͱ͠ɼ$͔Β"&ʹԼΖͨ͠ ਨઢͷΛ2ͱ͢Δͱɼ# 1 2ઢ্ʹ͋Δʯ
w ʮఆཧ ਖ਼ํܗ"#$%ͷล$%্ͷΛ&ͱ͢Δɽ&͔Β "$ʹԼΖͨ͠ਨઢͷΛ1ͱ͠ɼ$͔Β"&ʹԼΖͨ͠ ਨઢͷΛ2ͱ͢Δͱɼ# 1 2ઢ্ʹ͋Δʯ w Ծఆ ☐ABCD
C−D−E AC ⊥ EP A−C−P AE ⊥ CQ A−E−Q ݁ B−P−Q
w Ծఆ ☐ABCD C−D−E AC ⊥ EP A−C−P AE ⊥
CQ A−E−Q ݁ B−P−Q w *+ ࣗ༝ͳ ,+ ଞͷʹਵ ܾͯ͠·Δ (0, 0) (*1 , 0) (0, −*1 ) (*1 , −*1 ) (*1 , *2 ) (,3 , ,4 ) (,1 , ,2 )
w Ծఆ ☐ABCDʜࣗ໌ C−D−Eʜࣗ໌ AC ⊥ EP ʜ)* A−C−P ʜ)+
AE ⊥ CQ ʜ)- A−E−Q ʜ). ݁ B−P−Q ʜ/ 3FEVDFϓϩάϥϜ
! "#,…,"& 0 ͱͳͬͨɽ Αͬͯ ! ∈ )* , …
, )+ ఆཧূ໌͞Εͨɽ 3FEVDFϓϩάϥϜ࣮ߦ݁Ռ