U matematici rezultantom dvoh mnogochleniv P displaystyle P i Q displaystyle Q nad deyakim polem K displaystyle mathbb K zi starshimi koeficiyentami rivnimi odinici nazivayetsya viraz r e s P Q x y P x 0 Q y 0 x y displaystyle mathrm res P Q prod x y P x 0 Q y 0 x y inshimi slovami rezultant dorivnyuye dobutku poparnih riznic mizh yihnimi korenyami Dobutok beretsya za vsima korenyami v algebrichnomu zamikanni polya K displaystyle mathbb K z urahuvannyam yih kratnostej oskilki viraz sho vihodit ye simetrichnim mnogochlenom vid koreniv mnogochleniv P displaystyle P i Q displaystyle Q yaki mozhlivo ne nalezhat polyu K displaystyle mathbb K jogo mozhna zapisati yak mnogochlen vid koeficiyentiv P displaystyle P i Q displaystyle Q Dlya mnogochleniv starshi koeficiyenti yakih p displaystyle p i q displaystyle q vidpovidno ne obov yazkovo rivni 1 navedenij vishe viraz domnozhuyetsya na p d e g Q q d e g P displaystyle p degQ q degP Vlastivosti i sposobi obchislennya RedaguvatiOsnovna vlastivist rezultanta i jogo osnovne zastosuvannya rezultant mnogochlen vid koeficiyentiv P displaystyle P nbsp i Q displaystyle Q nbsp rivnij nulyu v tomu i lishe v tomu vipadku koli mnogochleni P displaystyle P nbsp i Q displaystyle Q nbsp mayut spilnij korin mozhlivo v deyakomu rozshirenni polya K displaystyle mathbb K nbsp Rezultant dorivnyuye viznachniku matrici Silvestra Diskriminant mnogochlena p mozhna viznachiti cherez rezultant p i jogo pohidnu p D p 1 1 2 n n 1 1 p n r e s p p displaystyle D p 1 frac 1 2 n n 1 frac 1 p n res p p quad nbsp de pn starshij koeficiyent mnogochlena p Dlya dovedennya spershu rozglyanemo vipadok pn 1 Todi mayemo g x x a i displaystyle g x prod x alpha i nbsp i pri x a j displaystyle x alpha j nbsp vikonuyetsya rivnist g x i j a j a i displaystyle g x prod i neq j alpha j alpha i nbsp Zvidsi oderzhuyemo r e s p p j p a j i j a j a i 1 1 2 n n 1 D p displaystyle res p p prod j p alpha j prod i neq j alpha j alpha i 1 frac 1 2 n n 1 D p nbsp Zvidsi j oderzhuyetsya chastkovij vipadok rivnyannya Zagalnij vipadok oderzhuyetsya yaksho vrahuvati sho pri domnozhenni mnogochlena p na konstantu pn rezultant res p p domnozhuyetsya na p2n 1 a diskriminant D p domnozhuyetsya na p2n 2 dd Rezultant rivnij dobutku znachen odnogo z mnogochleniv za korenyami inshogo yak i ranishe dobutok beretsya z urahuvannyam kratnosti koreniv r e s P Q P x 0 Q x displaystyle mathrm res P Q prod P x 0 Q x nbsp r e s P Q 1 deg P deg Q r e s Q P displaystyle mathrm res P Q 1 deg P cdot deg Q cdot mathrm res Q P nbsp r e s P R Q r e s P Q r e s R Q displaystyle mathrm res P cdot R Q mathrm res P Q cdot mathrm res R Q nbsp Yaksho P P R Q displaystyle P P R cdot Q nbsp i deg P deg P displaystyle deg P deg P nbsp todi r e s P Q r e s P Q displaystyle mathrm res P Q mathrm res P Q nbsp Yaksho X Y P Q displaystyle X Y P Q nbsp ye mnogochlenami odnakovih stepeniv i X a 00 P a 01 Q Y a 10 P a 11 Q displaystyle X a 00 cdot P a 01 cdot Q Y a 10 cdot P a 11 cdot Q nbsp todi r e s X Y det a 00 a 01 a 10 a 11 deg P r e s P Q displaystyle mathrm res X Y det begin pmatrix a 00 amp a 01 a 10 amp a 11 end pmatrix deg P cdot mathrm res P Q nbsp r e s P Q r e s Q P displaystyle mathrm res P Q mathrm res Q P nbsp de P z P z displaystyle P z P z nbsp Posilannya RedaguvatiStattya Weisstein Eric W Resultant na sajti MathWorld Otrimano z https uk wikipedia org w index php title Rezultant amp oldid 30536132
