🚀 go-pugleaf

In article <SHIGE.92Nov6112242@vega.tlab.yaskawa.co.jp>,
    shige@tlab.yaskawa.co.jp (Masaru Shigesada) writes:

shige> 	x^4 + x^3 == 0					(i)

shige>  x = u - 1/4$@$HCV$-49$($k!#(J

shige> 	u^4 + p u^3 + q u + r == 0			(ii)
		 $@",$3$3$O(J2$@$G$9$M(J$@!#(J
shige> 	p = -3/8, q = 1/8, r = -3/256			(iii)

shige> $@J,2rJ}Dx<0(J t^3 + 2 p t^2 + (p^2 - 4 r) t - q^2 == 0 $@$H$=$N:,$O!"(J

shige> 	t^3 - 3/4 t^2 + 3/16 t - 1/64 == 0		(iv)
shige> 			  (t - 1/4)^3 == 0
shige> 				    t == 1/4

$@$3$3$^$G$O$h$$$G$9$,!"(J

shige> $@$h$C$F!"(J(ii)$@$N:,$O!"(J

shige> 	u = 1/2 {sqrt(1/4) + sqrt(1/4) + sqrt(1/4)} = 3/4
shige> 	u = 1/2 {sqrt(1/4) - sqrt(1/4) - sqrt(1/4)} = -1/4 ... $@#3=E(J$@:,(J

$@$3$3$,BLL\$G$9!#855-;v$N(J

      $@$3$N4JLsJ}Dx<0$N:,$O!"$=$NJ,2rJ}Dx<0(J

	      t^3 + 2 p t^2 + (p^2 - 4 r) t - q^2 == 0

      $@$N#3:,(J t1, t2, t3 $@$r;H$C$F!"(J

	      u1,2 = (  Sqrt[t1] $@!^(J ( Sqrt[t2] + Sqrt[t3] ))/2
	      u3,4 = (- Sqrt[t1] $@!^(J ( Sqrt[t2] - Sqrt[t3] ))/2

      $@$N$h$&$KI=$9$3$H$,$G$-$k!#(J

$@$K$*$$$F!"(JSqrt[t?] $@$OJ#AG?t$N0UL#$NJ?J}:,$G$9$+$i!"(Jt? $@$,@5$N<B?t$@$+$i$H(J
$@$$$C$F(J Sqrt[t?] $@$,@5$G$"$k(J$@$H$$$&$o$1$G$O$"$j$^$;$s!#Be$o$j$K(J

	t1, t2, t3 $@$NJ?J}:,$NCf$G!"$=$N@Q$,(J -q $@$H$J$k$b$N$r(J
	Sqrt[t1], Sqrt[t2], Sqrt[t3] $@$H$9$k(J

$@$H$$$&$h$&$J>r7o$,$D$-$^$9!#(J

$@$3$NNc$G$9$H!"(Jt=1/4 $@$KBP$7!"$=$NJ?J}:,$G!"@Q$,(J -1/8 $@$K$J$k#3?t(J

	1/2, 1/2, -1/2

$@$rA*$S!"(J

	u_1 = ( 1/2 + ( 1/2 - 1/2 ) ) )/2 = 1/4
	u_2 = ( 1/2 - ( 1/2 - 1/2 ) ) )/2 = 1/4
	u_3 = (-1/2 - ( 1/2 + 1/2 ) ) )/2 = -3/4
	u_4 = (-1/2 + ( 1/2 - 1/2 ) ) )/2 = 1/4

(i)$@$N:,$O!"(J

	x = 1/4 - 1/4 = 0 ($@#3=E:,(J), -3/4 - 1/4 = -1

$@$G!"$a$G$?$/5a$^$j$^$9!#(J

 >>	   $@Lp%vIt(J $@4`(J: $@?t(JIII$@J}<0%,%m%"$NM}O@(J ---$@%"%$%G%"$NJQA+$r$a$0$C$F(J---
 >>		   $@8=Be?t3X<R(J (1976).

shige> $@$3(J$@$l$rFI$`$N$,@h$N$h$&$G$9$M!#(J

$@0J>e$O$9$Y$F$3$NK\$N<u$1Gd$j$G$9(J:-)
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$@2,It<wCK!w5~ETBg3X9)3XIt>pJs9)(J$@3X65<<7W;;5!%=%U%H%&%'%"9V:B(J