
thrikonaakrithiyilulla vighadanam

computer beejaganithathil, oru bahupada samvidaanathinte thrikonaakrithiyilulla vighadanam lalithamaaya bahupada samvidaanangalude oru koottamaanu, athaayath oru pooyint sistangalilonninte parihaaramaanengil maathram athinte parihaaramaan.
athinte coffificient feeldinte beejaganita klosaril parihara settu vivarikkuka ennathaanu uddeshyam, aaa lalithamaaya samvidaanangal saadhaarana shringalakalaanu. bahupada samvidaanangalude gunakangal yadhaartha samkhyakalaanenkil, thrikonaakrithiyilulla vighadippichu saadhaarana ardha-beejaganita samvidaanangalaayi athinte yadhaartha pariharangal labikkum. rund saahacharyangalilum, eee lalithamaaya samvidaanangalkku oronninum thrikonaakrithiyum sradheyamaaya gunangalumundu, ithu padawali nyaayeekarikkunnu.
charithram.
oru beejaganita vaividhyathe thulyamaana gadakangalaayi vighadippikkunnathinu nirddeshikkappetta aadyathe factorisation free algorithamaanu characteristic settu reethi. koodaathe, ezhuthukaaranaaya wenn-sann vuu eee reethi nadappaakkunnathu manassilakkukayum 1987 leeii thante pioneer lekhanathil "polynomial samavaakyangal pariharikkunnathinulla oru poojya ghadana sidhaantham" enna thalakkettil pareekshanathmaka daatta report cheyyukayum cheythu. eee kruthi sandarbhathilekku konduvarunnathinu, eee lekhanam ezhuthiya samayat oru beejaganita settu vighadanathekkurichulla pothuvaaya ashayam enthaayirunnuvennu namukk oorkkaam.
beejaganithaparamaayi adacha feeld aayirikkukayum oru subfield aayirikkukayum cheyyuka. poojyangalude koottam thulyamaaya oru polynomial settu undengil oru upavibhagam oru (affine) beejaganita vakabhedamaanu.
ellah beejaganita enangalkkum bandham soochippikkunnathu onno allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allengil allenki aadyathe beejaganita vaividhyam vighadanam prasidhamaaya lasker-nother sidhaanthamaanu, athu inipparayunnava soochippikkunnu.
melpparanja sidhaanthathile enangale irreducible gadakangal ennu vilikkunnu, avaye oru vighadana algorithathinte swaabhaavika auttputtaayi kanakkakkaam, allengil mattoru vidhathil paranjaal, oru samavaakyangalude sisttam pariharikkunna oru algoritam.
oru computer prograamilekku nayikkunnathinu, eee algoritam specification irreducible gadakangal engane prathinidheekarikkunnuvennu nirddeshikkanam. atharamoru encoding thaazhepparayunna phalatthiloode joseph ritt avatharippikkunnu.
ritt sidhaanthathile settine nammal ideal formula _ 2-nte ritt characteristic settu ennu vilikkunnu. thrikonaakrithiyilulla settu enna aashayathinaayi dayavaayi regular cheyin nokkuka.
feeld extensionukalkku mukalilulla polynomial factorisation, prime idealukalude characteristic settukalude kanakkukoottal ennivaye adisthaanamaakkiyulla polynomial sisttangal pariharikkunnathinulla oru reethi joseph ritt vivarichu.
ennirunnaalum, eee reethi praayogikamaayi nadappaakkunnathu budhimuttulla oru prashnamaayirunnu, ippozhum nilanilkkunnu. 1980kalil, caractoric settu reethi avatharippichappol, polynomial factorisation oru sajeeva gaveshana mekhalayaayirunnu, eee vishayathekkurichulla chila adisthaana chodyangal aduthide pariharichu.
innathe kaalath, mikka prayoga prashnangalum prosess cheiyunnathinu oru beejaganita inathe irreducible gadakangalaayi vighadippikkendathu aavasyamilla, kaaranam vighadippikkalinekkuri durbalamaaya ashayangal, kanakkukoottaan chelavu kuravaanu, mathiyakum.
swabhaavasavisheshathakalude gana reethi ritt sidhaanthathinte thaazhepparayunna vakabhedathe aasrayichirikkunnu.
vyathyastha ashayangalum algorithangalum wenn-sann vuu-yude pravartthanam vipuleekarichu. 1990 kalude thudakkathil, 1991 ll maikkal calcbrenner thante phd theesisil swathanthramaayi avatharippicha oru saadhaarana shringhala enna ashayam, luu yangum jingshong shang chernnu pradhanappetta algoritam kandethalukalilekku nayichu.
calcbrenarude kaazhchayil, oru beejaganita vaividhyathinte irreducible gadakangalude pothuvaaya poojyangale prathinideekarikkaan saadhaarana shringalakal upayogikkunnu. yaanginteyum shaanginteyum yadhaartha kruthiyil, oru hypersurfus oru ardha-vakabhedathe (oru saadhaarana shringhala nalkunna) murichuvidunnundo ennu theerumaanikkaan avaa upayogikkunnu. saadhaarana shringalakalkku vaasthavathil niravadhi rasakaramaaya gunangalundu, beejaganita allengil deferential samavaakyangalude sisttangale vighadippikkunnathinulla pala algorithangalilum ithu pradhana aashayamaanu.
pala pathrangalilum saadhaarana shringalakal anweshikkappettittund.
eee vishayathekkurichulla samriddhamaaya saahityam oru saadhaarana shringalayude niravadhi thulyamaaya nirvachanangaliloode vishadeekarikkaanaakum. yadhaarthathil, kaalkbrenarinte yadhaartha roopeekaranam yangilum shangilum ninnu thikachum vyathyasthamaan. eee rund aashayangalkkidayilulla oru paalamaya calcbrenarinteyum yaanginteyum shanginteyum kaazhchappaad dongming vaanginte kadalaasil prathyakshappedunnu.
calcbriner enna ardhathilum lazard, wenn-sann vuu ennee ardhathilum thrikonaakrithiyilulla vighadanam labhikkunnathin vividha algorithangal labhyamaan. daniel lasardinte lextriangular algorithavum maarkk moreno maasayude triade algorithavum characteristic settu reethiyum axium, maple ennivayulppede vividha computer aljibra sisttangalil labhyamaan.
aupachaarika nirvachanangal.
oru feeld ayirikkatte, variableukal oordar cheyyappedatte. anubandha polynomial ring upayogichaanu nammal soochippikkunnathu. kaaranam, bahupada samavaakyangalude oru samvidaanamaayi kanakkakkappedunnathinaal, beejaganita klosaril oru thrikonaakrithiyilulla vighadanathekkurichu rund aashayangalund. aadyathethu, calcbriner ennu vilikkappedunna ardhathil beejaganita settinte pothuvaaya pooyintukal maathram prathinidheekarichu alasathayode vighadippikkuka ennathaanu.
randaamathethu, lazard, wenn-sann vuu ennee ardhangalil vilikkappedunna ellah pooyintukalum vyakthamaayi vivarikkuka ennathaanu.
rund saahacharyangalilum parimithamaaya niravadhi saadhaarana shringalakalundu, formula _ 8 pooritha aidiyalinte radicaline soochippikkunnu, athesamayam athinte ardha gadakathe soochippikkunnu. eee aashayangalude nirvachanangalkkayi dayavaayi saadhaarana shringhala kaanuka.
inii muthal athu oru yadhaartha clossed feeldaanennu karuthuka. bahupadangalulla oru ardha beejaganita samvidaanam pariganikkuka. namukk ullathupole parimithamaaya niravadhi saadhaarana ardha-beejaganita samvidaanangal nilavilundu.
evideyaanu pariharamundaakkunna pooyintukale soochippikkunnathu. saadhaarana ardha-beejaganita samvidaanangal ardha-beejaganita samvidaanathinte thrikonaakrithiyilulla vighadanam undaakkunnu.
udaaharanagal.
yukthisahamaya samkhya feeldine soochippikkuka. variable oordaring formula 11 ulla formula _ 10 ll, inipparayunna polynomial sisttam pariganikkuka 
maple coode anusaricha 
(regular cheyinukal):
r: = polynomialring ([x, y, z]):
sys: = {x ^ 2 + y + z-1, x + y ^ 2 + z-1, x + y + z ^ 2-1}:
l: = thrikonaakrithiyilulla (sys, r):
bhoopadam (samavaakyangal, l, r);
regular cheyin library upayogichu laayaniyude saadhyamaaya thrikonaakrithiyilulla vighadanam ithaanu

