bahugunitata (ganeet)

ganitat, bahusanchachya sadasyachi bahusankhya mhanje bahusanchachya sadasyamadhye tee kitee vela disate. udaharanarth, dilelya bahupadila dilelya binduvar kitee vela mool aahe, tee tyaa moodachi gunakar aahe.
apavad nirdisht naan karata yogyaritya mojanyas saksham honyasathi (udaharanarth, donada mojali jaanari 'duheri mule') bahuvishayachi kalpana mahattvachi aahe. mhanoon abhivyakti, "bahuvidhatene ganali jaate".
jarr bahuvidhatatekade durlaksh kele gele tarr 'bhinn moolanchi sankhya' pramane 'vishisht' ghatakanchi sankhya mojoon yavar bharr deela jaau shakato. tathapi, jevha jevha ekhada sanch tayar hoto (bahusanchachya ulat), tevha 'vishisht' haah shabd naan vaparta, bahugunatekade aapoaap durlaksh kele jaate.
mukhya ghatakachi gunakar.
abhajya ghatkamadhye, abhajya ghatakachi gunakarata heey tyache sootra _ 1-edic moolyankan asate. udaharanarth, purnankache mukhya ghatakatva aahe
mukhya ghatakachi gunakarakta aahe, tarr pratyek mukhya ghatakachi gunakarata aahe aanee aahe. asha prakare, gunakaras parvanagi denare chaar mukhya ghatak aahet, parantu keval teen bhinn mukhya ghatak aahet.
bahupadichya moodachi gunakar.
sootra _ 2 heey eka kshetra asoo dya aanee sootra _ 3 heey sootra _ 2 madheel gunank asalelya eka chalamadhye bahupadi asoo dya. jarr sootra _ 9 aanee sootra _ 10 asey bahupadi sootra _ 8 asel tarr ghatak sootra _ 5 heey sootra _ 3 chya gunakar sootra _ 6 chey mool aahe. jarr sootra _ 11 asel tarr 'a' laa sadhe mool mhanatat. jarr sootra _ 12 asel tarr sootra _ 13 laa ekadhik mool mhanatat.
udaharanarth, bahupadi sootra _ 14 madhye 1 aanee −4 heey mool aahet aanee teey sootra _ 15 mhanoon lihile jaau shakate. yacha arth assa qee 1 heey gunakar 2 chey mool aahe aanee −4 heey eka sadhe mool aahe (gunakar 1 chey). bijganitechya moolbhut prameyadvare bahupadichya sampoorn gunankanat yaa moodachya ghatananchi sankhya hii moodachi gunakar aahe.
jarr sootra _ 13 heey bahupadichya gunakar sootra _ 6 chey mool asel, tarr teey tyaa bahupadichya vyutpattichya gunakar sootra _ 18 chey mool aahe, joparyant antarnihit kshetrache vaishishtya vibhajak nasate, asha paristhitit sootra _ 13 heey gunakarche mool aahe, kiman vyutpattiche sootra _ 6.
bahupadiche jarr aanee jarr bahupadiche mool ekadhik asel tarach bahupadicha bhedbhav shoonya asato.
bahuvidh moodajaval bahupadi karyache vartan.
bahupadi karya 'f' chaa aalekh bahupadichya vastavik mulanvar 'x'-akshala sparsh karato. aalekh haah 'eff' chya anek mulanvar sparshika asato aanee sadhya mulanvar sparshika nasato. aalekh visham bahutechya mulanvar 'x'-aksh olandato aanee summ bahutechya mulanvar tyala olandat nahee.
jarr aanee keval tyachya serve mulanmadhye summ gunakar asel aanee sootra _ 21 asey sootra astitvat asel tarr shoonya nasalele bahupadi karya sarvatra nakaratmak nasate.
samikarananchya araikhik pranalichya samadhanachi gunakar.
ekal chull samadhan sootra _ 24 asalelya samikaran sootra _ 23 sathee, gunakar sootra _ 6 asel, jarr
jarr shabdant, vibhedak karyatmak sootra _ 28, sootra _ 24 varr karyache vyutpann sootra _ 29 mhanoon paribhashit kelele, sootra _ 32 sathee sootra _ 31 varr sootra _ 18 paryant nahise hote. teey vibhedak karyatmak sootra _ 34 vector spacemadhye pasaratat, jyala sootra _ 24 varr mccaulay dual space mhanatat aanee tyache pariman sootra _ 24 chey gunakar aahe, sootra _ 31 chey shoonya mhanoon.
sootra 38 hii sootra 40 chalanchi sootra 39 samikarananchi eka pranali asoo dya, jyamadhye sootra 42 heey sootra 43 teey sootra 44 kinwa sootra 45 teey sootra 46 paryantche mapping aahe. sootra 41 varr vibhedak karyanchi eka mccaulay duheri jaaga dekhil aahe jyamadhye pratyek karya sootra 42 varr nahishi hote. yaa mccaulay duheri jaageche pariman samikaran sootra 38 chya samadhan sootra 41 chee gunakar aahe. mccaulay duheri jaaga samadhanavar pranalichi gunakar rachana tayar karate.
udaharanarth, samikarananchya pranaliche sootra _ 51 heey sootra _ 38 chya swarupat
gunakar 3 aahe kaaran macauli duheri jaaga aahe 
heey pariman 3 chey aahe, jethe sootra _ 55 heey bindu sootra _ 51 varr karyavar lagoo kelele vibhedak karyatmak sootra _ 56 darshavate.
jarr samadhan vegale asel tarr gunakar nehmich maryadit asato, yaa arthane vikshobh aparivartaniy aahe qee sootra _ 6-putt samadhan heey jatil jaganmadhil vikshobhakhali sanyukt gunakar sootra _ 6 saha samadhanancha samooh banate aanee bahupadi pranalinvaril chhedanbindu gunakarasarkhech asate.
aantarchhedan bahugunitata.
bijganitiy bhoomitimadhye, bijganitiy vividhatechya done up-prakaranche chhedanbindu heey aparivartaniy prakaranche maryadit sanyojan aahe. asha chhedanbinduchya pratyek ghatkala eka 'chhedanbinduchi bahulata' jodali jaate. hii kalpana yaa arthane sthanik aahe qee yaa ghatakachya konatyahi samanya binduchya parisarat kaay ghadate teey pahoon tyachi vyakhya kelee jaau shakate. asey aahe qee, vyapakata naan gamavata, aapan aantarchhedan bahugunitata paribhashit karnyasathi, done sanlagn prakaranche (sanlagn jaageche up-prakar) chhedanbindu vicharat gheu shakato.
asha prakare, 'v' 1 aanee 'v' 2 yaa done sanlagn prakaranna pahata, 'v' 1 aanee 'v' 2 chya chhedanbinduchya 'w' yaa aparivartaniy ghatakacha vichar qaraa. 'd' haah 'w' chaa pariman asoo dya aanee 'p' haah 'w' chaa konatahi samanya bindu asoo dya. p" madhoon madhoon samanya sthiteet "d" hyperplanes "w" chya chhedanbindumadhye eka irreducible ghatak asato joou ekach binduvar "p" paryant qamee hoto. mhanoon, chhedanbinduchya nirdeshank ringchya yaa ghatkavaril sthanik ringla fakt eka avibhajya aadarsh aahe aanee mhanoon tee eka artinian ring aahe. asha prakare hii ring jaminivaril kshetravaril eka maryadit aayami vector space aahe. tyache pariman mhanje 'v' 1 aanee 'v' 2 chee 'w' varchi aantarchhed gunakar.
hii vyakhya aaplyala bezoutche pramey aanee tyache samanyikaran tantotant sanganyas anumati dete.
hii vyakhya bahupadichya moodachya gunakarala khaleel prakare samanya karate. bahupadi "f" chee mule heey sanlagn reshevaril bindu asatat, jeey bahupadidvare paribhashit bijganitiy sanchache ghatak asatat. yaa sanlagn sanchatil nirdeshank ring sootra _ 60 aahe jethe "keey" heey "eff" chey gunank asalele bijganitiyadrushtya bund kshetra aahe. jarr sootra _ 61 heey "f" chey gunank asel, tarr mukhya aadarsh sootra _ 62 vareel "r" chee sthanik ring sootra _ 63 aahe, hii "k" vareel vector space aahe, jyat moolche gunakar sootra _ 64 pariman mhanoon aahe.
aantarchhedan bahugunitatechi hii vyakhya, jee moolatah jeane-piyare sermule tyachya 'lokal bijganit' yaa pustakat aahe, tee keval aantarchhedan kelelya sanch saiddhantik ghatkansathi (jyala 'isolated components' asehi mhanatat) karya karate, antarbhut ghatkansathi nahee. antarbhut prakaran hatalanyasathi siddhant vikasit kele gele aahet (tapashilansathi chhedanbindu siddhant pahaa).
jatil vishleshan.
z" 0 heey holomorphic function "f" chey mool asoo dya aanee "n" heey sarvat qamee sakaratmak purnank asoo dya jenekarun "z" 0 varr moolyankan kelele "f" chey "n" weey vyutpann shoonyapeksha vegale asel. mug 'zed' 0 chya aaspas 'eff' chee shakti malika 'aine' vya sandnyepasun suru hote aanee 'eff' laa gunakar (kinwa 'kram') 'aine' chey mool asalyache mhatale jaate. jarr "n" = 1 asel tarr moodala sadhe mool mhanatat.
aapan meromorphic functionschya shoonya aanee dhruvanchi gunakar dekhil paribhashit karoo shakato. jarr aaplyakade eka meromorphic function sootra _ 65 asel tarr "g" aanee "h" chey teller vistar sumare eka bindu "z" 0 ghya aanee pratyekamadhye pahili shunyetar sandnya shodha (anukrame "m" aanee "n" yaa sandnyancha kram darshava) tarr jarr "m" = "n" asel tarr binduche moolya shunyetar aahe. jarr sootra _ 66 asel tarr bindu gunakar sootra _ 67 chaa shoonya asel jarr sootra _ 68 asel tarr bindula gunakar sootra _ 69 chaa dhruv aahe.

