🚀 go-pugleaf

In article <SHIGE.92Nov4182038@siva.tlab.yaskawa.co.jp>,
    shige@tlab.yaskawa.co.jp (Masaru Shigesada) writes:

shige>   $@#4<!$NBe?tJ}Dx<0(J$@$K4X$9$k<ALd$G$9(J ^o^;$@!#(J
shige> ---
shige> 	a x^4 + b x^3 + c x^2 + d x + e == 0

shige> $@$r2r$/>l9g!"$^(J$@$:(J x = u - b/(4 a) $@$HCV$-49$($F4JLs7A(J

	u^4 + p u^2 + q u + r == 0

shige> $@$KD>$9!#(J

($@CfN,(J)

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

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

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

$@!V%*%$%i!<$N2r(J$@K!!W$H8F$P$l$F$$$kJ}K!$G$9$M!#(J

shige> 	x^4 + x^3 == 0

shige> $@$r2r$/$H!"(Jx = 1/2, -1/2$@!J#3=E2r!K$K$J$C(J$@$F$7$^$$$^$9!#(J
                 ~~~~~~~~~~~~~~~~~~~~~~
$@$3$N7k2L$,=P$kF;6Z$r<($7$FD:$1$^$9$+!)(J

shige> ii)$@FI$s$@$[$&$,$$$$=q@R(J

$@$H$7$F$O!"8=:_F~<j2DG=$+$I$&$+$OCN$j$^$;$s$,!"(J

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

$@$r$*4+$a$7$F$*$-$^$9!#(J

shige> $@<B?t78?t$N(J$@#4<!J}Dx<0(J(a$@!b(J0)$@$K$D$$$F!"J#AG?t$r;H$o$:$K!";MB'1i;;$H;03Q4X(J
shige> $@?t!&;X?t4X?t!&(Jsqrt[2]{ }$@!&(Jsqrt[3]{ }$@$r;H$C$F!"$9$Y$F$N<B?t2r$r5a$a$k$3(J
shige> $@$H$rL\I8$H$7$F$$$^$9!#(J

$@$3$NL\I8$N$?$a$J$i!"2<5-$N!V%U%'(J$@%i%j$N2rK!!W$NJ}$,E,$7$F$$$k$+$b$7$l$^$;$s!#(J

$@4JLs7A(J

	u^4 + p u^2 + q u + r = 0

$@$KBP$7!"#3<!J}Dx<0(J

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

$@$r9M$($k!#$3$N#3<!J}Dx<0$N0l$D$N:,$r(J t_0 $@$H$9$k$H!"(J
t_0 \neq p $@$N$H$-!"4JLs7A#4<!J}Dx<0$O!"F1CM$JJ}Dx<0(J

    ( u^2 + t_0 / 2 )^2 = \{ \sqrt{t_0-p} u - q / ( 2 \sqrt{t_0-p} ) \}

$@$KJQ7A$7$F2r$1$k!#(Jt_0 = p $@$N$H$-$O(J q = 0 $@$G(J

	u^4 + p u^2 + r = 0

$@$G2r$1$k!#(J($@>e5-(J$@J88%;2>H(J)
-----
$@2,It<wCK!w5~ETBg3X9)3XIt>pJs9)3X65<<7W;;5!%=%U%H%&%'%"9V:B(J