A.5.2 $B%f!<%/%j%C%I$N8_=|K!$K$h$k(B Strum $BNs$N@8@.(B

$BB?9`<0(B $f(x)\in\R[x]$ $B$,M?$($i$l$?$H$-!"(B $f_0(x)=f(x)$ $B$H(B $f_1(x)=f'(x)$ $B$+$i(B Euclid $B$N8_=|K!$r9T$$!"4X?tNs(B $f_0(x)$, $f_1(x)$, $\cdots$, $f_\ell(x)$ $B$r:n$k(B:

$$f_0(x):= f(x), \quad f_1(x):= f'(x),$$ (A.5)

$$f(x) = q_1(x)f_1(x)-f_2(x), \quad \deg f_{2}(x)<\deg f_{1}(x),$$

$$f_1(x) = q_2(x)f_2(x)-f_3(x), \quad \deg f_{3}(x)<\deg f_{2}(x),$$

$$f_2(x) = q_3(x)f_3(x)-f_4(x), \quad \deg f_{4}(x)<\deg f_{3}(x),$$

$$\vdots$$

$$f_{k-1}(x) = q_{k-1}(x)f_{k}(x)-f_{k+1}(x), \quad \deg f_{k+1}(x)<\deg f_{k}(x),$$ (A.6)

$$\vdots$$

$$f_{\ell-2}(x) = q_{\ell-1}(x)f_{\ell-1}(x)-f_\ell(x), \quad \deg f_{\ell}(x)<\deg f_{\ell-1}(x),$$

$$f_{\ell-1}(x) = q_{\ell}(x)f_\ell(x).$$

($BIaDL$N8_=|K!$H0[$J$j!"(B $f_{k-1}(x)$ $B$r(B $f_{k}(x)$ $B$G3d$C$?$H$-$N(B $BIaDL$N>jM>(B $\times (-1)$ $B$r(B $f_{k+1}(x)$ $B$H$9$k!#(B)

$B$h$/CN$i$l$F$$$k$h$&$K(B $f_\ell(x)$ $B$O(B $f(x)$ $B$H(B $f'(x)$ $B$N:GBg8xLsB?9`(B $B<0$G$"$k$+$i!"(B $f(x)\in\R[x]$ $B$,=E:,$r;}$?$J$$>l9g!"(B $f_\ell(x)\equiv$   $BDj?t(B ($\ne 0$) $B$H$J$k$3$H$KCm0U$7$h$&!#(B $B0J2<$3$N>l9g$K(B $f(x)=0$ $B$N2r(B ($B:,(B) $B$r5a$a$k$3$H$r9M$($k!#(B $f(x)$ $B$,=E:,$r;}$D>l9g$O(B $f(x)$ $B$NBe$o$j$K(B $g(x)=f(x)/f_\ell(x)$ $B$r9M$($k$3$H$GF1MM$N5DO@$,$G$-$k!#(B

Proof. $B!"!+!"%3(B $f_\ell(x)\equiv$$B%HtA$G$"$k$+$i!"(BStrum $BNs$N>r7o(B (3) $B$OK~(B $B$?$5$l$F$$$k!# $x_0\in [a,b]$, $B$"$k(B $k\in\{0,1,\cdots,\ell-1\}$ $B$KBP$7$F(B

$$f_{k}(x_0)=f_{k+1}(x_0)=0$$

$B$H$J$C$?$H$9$k$H!"<0(B (A.8) $B$+$i(B $f_{k+2}(x_0)$ $B0J9_$N(B $f_j(x_0)$ $B$b$9$Y$F(B 0 $B$K$J$j!"FC$K(B $f_\ell(x_0)=0$. $B$3$l$O(B $f_\ell(x)$ $B$,Dj?t4X?t(B ($\ne 0$) $B$G$"$k$3$H$KL7=b$9$k!#$f$($K(B Strum $BNs$N>r7o(B (1) $B$,(B $BK~$?$5$l$k!#(B $B, $B$"$k(B $k\in\{1,2,\cdots,\ell-1\}$ $B$K(B $BBP$7$F(B $f_k(x_0)=0$ $B$H$J$C$?$H$9$k$H!"<0(B (A.8) $B$+$i(B

$$f_{k-1}(x_0)=-f_{k+1}(x_0).$$

$B$f$($K(B Strum $BNs$N>r7o(B (2) $B$bK~$?$5$l$k(B ($B>r7o(B (3) $B$+$i>e<0$NCM$O(B 0 $B$K$J(B $B$i$J$$$3$H$KCm0U(B)$B!#(B $B:G8e$K(B $x_0$ $B$,(B $f(x)$ $B$N:,$G$"$k$H$-!"(B$f(x)$ $B$,=E:,$r;}$?$J$$$H$$$&(B $B2>Dj$+$i(B $f'(x_0)\ne 0$ $B$G!"(B

$$f'(x_0)f_1(x_0)=f'(x_0)^2>0$$

$B$H$J$j!">r7o(B (4) $B$bK~$?$5$l$k!#(B $\qedsymbol$ ARRAY(0x12f60a8) $\qedsymbol$