ERDŐS YA TAMBAYA A 1961. RABI, KUSAN BABU SHAKKA

Jefa tsabar kuɗi ga kowane coefficient na polynomial: kai yana ba da +1, wutsiya tana ba da −1. Sakamakon shi ne polynomial ta Littlewood ta bazuwar (random Littlewood polynomial), 1 ± z ± z² ± … ± zⁿ. Kamar kowace polynomial mai daraja n, tana da tushe n — wurare a cikin jirgin lambobi masu sarƙaƙiya (complex plane) inda take daidai da sifili.

A 1961, masanin lissafi Erdős ya tambayi ina waɗannan tushen ke zuwa. A taƙaice: yayin da daraja ke girma, shin juzu’in tushen da ke cikin rufaffen faifan raka’a — da’irar mai radius 1 da duk abin da ke cikinta — yana kusantar daidai rabi, da yiwuwa ta ɗaya? Tambayar ta zama Matsala ta #522 a cikin jerin matsalolin Erdős. Wani sigar mai rauni, gaskiya “da yiwuwa mai girma” maimakon “kusan babu shakka”, Yakir ya tabbatar da ita a baya.

Sebastien Kawada na MIT yanzu ya amsa tambayar Erdős: eh.

Manne da da’irar

Takardar ta tabbatar da fiye da rabin. Kusan dukkan tushen suna cunkushewa a cikin wani siririn ɗamara a kewayen da’irar raka’a, a nisa mai girman 1/n. Idan aka zuƙo wannan ɗamarar da canji x = n(|z| − 1), rabon tushen da ke cikin da’irar mai radius 1 + x/n yana daidaitawa a kan lanƙwasa ɗaya:

Φ(x) = ½ (1 + coth x − 1/x)

A x = 0, lanƙwasar tana ba da daidai ½: amsar Erdős. Wannan lanƙwasar tana aiki ga dukkan iyalin coefficient na bazuwar — jefa tsabar kuɗi, ɗigogi na bazuwar a kan da’ira, lambobin Gauss. Wannan shi ne ɓangaren gama-gari (universal) na taken.

Sakamakon ma yana aiki idan polynomial masu daraja daban-daban suna da alaƙa ta kowace hanya: an gina su ta ƙara jefa tsabar kuɗi ɗaya bayan ɗaya, ko an zana su sabo gaba ɗaya a kowace daraja. Ga jefa tsabar kuɗi, yawan tushen da ke cikin faifan fiye da n/2 yana zama ƙasa da kusan n^(3/4).

Inda gama-garin ke tsayawa

Daga nan takardar ta duba ƙananan sauye-sauye a kewayen rabi. A nan dokar rarraba coefficient ta fara da muhimmanci. Ga coefficient na Gauss, bambancin (variance) yawan rarar yana girma kamar 0.3059 × n. Ga wasu dokokin rarraba masu daidaito, yana sauyawa da kashi ɗaya bisa goma sha biyu na wani adadi da ake kira fourth cumulant (cumulant na huɗu), wanda ke auna nauyin wutsiyoyin dokar. Ga jefa tsabar kuɗi, adadin ya zama 0.3059 − 1/6. Gwaje-gwajen kwamfuta a kan polynomial na bazuwar 16,384 masu daraja har zuwa miliyan ɗaya sun yi daidai da waɗannan hasashe.

Yadda aka tabbatar da shi

Kayan aiki biyu na gargajiya ne ke yin babban aikin. Dabarar Jensen (Jensen’s formula) tana mayar da ƙidayar tushe a cikin faifai zuwa matsakaicin logarithm na polynomial a kewayen da’ira; wannan matsakaici ya zama yana taruwa sosai. Mataki mai wuya shi ne tafiya daga jerin darajoji masu warwatse zuwa kowace daraja a lokaci guda. Don sauye-sauyen, ƙa’idar argument (argument principle) tana mayar da rarar tushen zuwa ƙidaya mai alama ta sau nawa wani lanƙwasa, da polynomial ke zanawa a kan da’irar raka’a, ke ketare wani layi.

Na’ura ta tabbatar

Kowane babban sakamako na takardar mai shafi 82 an tabbatar da shi a tsari a cikin mataimakin hujja na Lean 4. An karɓi hujjar Matsala ta #522 a matsayin warwarewa ga bayaninta na tsari a cikin aikin Formal Conjectures na Google DeepMind, kuma fayil ɗaya yana bai wa kowa damar sake gudanar da ita a cikin burauzar yanar gizo. Marubucin ya rubuta cewa tsarin AI na cikin gida daga ɗakin bincike na Manolis Kellis a MIT, ciki har da tsarin harshe, sun taimaka wajen tsara Lean da duba wasu lissafe-lissafe; marubucin ya bayyana cewa hanyar lissafin tasa ce.

A watan Afrilu 2026, an wallafa hujjoji biyu marasa tsari na wannan bayanin a dandalin tattaunawa na matsalolin Erdős. Kawada ya ce wannan hujja da hanyarta na asali ne, kuma ita ce ta farko da aka tabbatar a tsari. Shekaru sittin da biyar bayan an yi tambayar, amsar da na’ura ta tabbatar ta ce: rabi, kusan babu shakka.