Naoya Umezaki
October 06, 2018
650

# 整数論と様々な数学

MATHPOWER2018での講演。フィールズ賞受賞者Akshay Venkateshの業績紹介。

October 06, 2018

## Transcript

3. ### ೋ࣍ܗࣜ ϥάϥϯδϡͷ࢛ฏํ࿨ఆཧ x2 + y2 + z2 + w2 Ͱશͯͷ੔਺Λද͢ɻ

10 = 12 + 32 15 = 32 + 22 + 12 + 12
4. ### ೋ࣍ܗࣜ ೋ࣍ܗࣜͷม׵ P(x1 , x2 , x3 ) = x2

1 + x2 2 + x2 3 Q(y1 , y2 ) = 2y2 1 + 2y1 y2 + 2y2 2 Λߟ͑Δɻ x1 = y1 + y2 , x2 = y1 , x3 = y2 ͱ͢Δɻ
5. ### ೋ࣍ܗࣜ P(x1 , x2 , x3 ) = x2 1

+ x2 2 + x2 3 Q(y1 , y2 ) = 2y2 1 + 2y1 y2 + 2y2 2 P(x1 , x2 , x3 ) = (y1 + y2 )2 + y2 1 + y2 2 = 2y2 1 + 2y1 y2 + 2y2 2
6. ### ೋ࣍ܗࣜ ͋Δೋ࣍ܗࣜQ ͕ɺଞͷೋ࣍ܗࣜP ͔Βม਺ ม׵ͰදݱͰ͖Δ͔ʁmม਺ͷP ͕nม਺ͷ Q Λදݱ͢Δ͔ʁ ہॴେҬݪཧʢϋοηݪཧʣ p

ਐ਺Qp ͷൣғͱ࣮਺RͷൣғͰߟ͑Δɻ શͯͷp ٴͼRͰදݱͰ͖Ε͹༗ཧ਺ͷൣғ Ͱ΋දݱͰ͖Δ͔ʁ

8. ### ϦχοΫ༧૝ ੪࣍ଟ߲ࣜQ ʹର͠ɺQ(x) = d ͳΔx ͷू ߹ɻd ͰׂͬͯɺQ(x) =

1Ͱͷd → ∞Ͱͷ ෼෍ͷ༷ࢠɻ Q(x) = x2 1 + x2 2 + · · · + x2 n Λߟ͑Δͱɺٿ໘্ ͷ༗ཧ఺ͷ෼෍ɻ ܈ͷ࡞༻͕͋Δ৔߹Λߟ͑Δɻௐ࿨ղੳͱΤ ϧΰʔυཧ࿦Λ࢖͏ɻ
9. ### ΠσΞϧྨ܈ͷ෼෍ ΠσΞϧྨ܈ͱ͸ʁ੔਺ͰͷૉҼ਺෼ղͷҰ ҙੑ 6 = 2 × 3 10 =

2 × 5 √ −5Λ෇͚Ճ͑Δͱ่ΕΔ 6 = 2 × 3 = (1 + √ −5)(1 − √ −5)

12. ### ΠσΞϧྨ܈ͷ෼෍ ΤϨϯόʔά-ϰΣϯΧςγϡ-΢Σε λʔϥϯυ ίʔΤϯɺϨϯετϥ༧૝ͷؔ਺ମྨࣅΛূ ໌ͨ͠ɻ ؔ਺ମFp (x, a)͸༗ݶମ্ͷ୅਺ۂઢͷ༗ཧ ؔ਺શͯूΊͨ΋ͷɻ͜Ε΋ಉ༷ʹΠσΞϧ ྨ܈ΛఆٛͰ͖Δɻ

ϑϧϏοπۭؒͷϗϞϩδʔ҆ఆੑΛ࢖ͬͯ