OpenXM/doc/compalg/factor.tex - diff

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version 1.1.1.1, 2000/03/01 02:25:51

version 1.2, 2000/03/28 01:59:21

Line 1216 F(x) &=& x^{16} (x^2-28)^8 (x^2-20)^8 (x^2-8)^8 (x^2-1

$B0lHL$K(B, $BM-M}4X?t(B $f(x)=n(x)/d(x)$ ($n,d \in \Q[x]$) $B$NITDj@QJ,$O(B,

$f$ $B$NItJ,J,?tJ,2r$K$h$j7W;;$G$-$k(B. $B$7$+$7(B, $B$=$N$?$a$KJ,Jl(B $d$ $B$r(B

1 $B<!0x;R$N@Q$KJ,2r$9$k$K$O(B $d$ $B$N:G>.J,2rBN$r5a$a$k$3$H$,I,MW$H$J$k(B.

$B$^$?(B, $BITDj@QJ,<+BN$K(B, $d$ $B$NJ,2r$K$h$j8=$l$?Be?tE*?t$,8=$l$k$H$O(B

$B8B$i$J$$(B.

\begin{ex}

Line 1290 $$\int f dx = \sum_i c_i\log r_i$$

Line 1289 $$\int f dx = \sum_i c_i\log r_i$$

$B$H=q$1$k(B. $B0lHL$K(B $c_i \in \Q, r_i \in \Q[x]$ $B$H$O8B$i$:(B, $B2?$i$+$NBe?tE*?t$r4^$`(B

$B2DG=@-$,$"$k$,(B, $B$3$NBe?t3HBg$r:G>.8B$K$9$k$h$&$JI=<($r5a$a$?$$(B.

\begin{pr}(Rothstein)

\begin{pr}(Rothstein{\rm\cite{DAV}})

$K$ $B$rJ#AG?tBN(B $\C$ $B$NItJ,BN$H$7(B,

$f(x)=n(x)/d(x)$, $n,d \in K[x]$, $\GCD(n,d)=1$, $\deg(n)<\deg(d)$ $B$G(B $d$ $B$OL5J?J}(B, $BL5J?J}$H$9$k(B. $B$3$N$H$-(B,

$$n/d = \sum_{i=1}^n c_i v_i'/v_i$$

$B$?$@$7(B $c_i \in \C$ $B$OAj0[$j(B, $v_i \in \C[x]$, $v_i$ $B$O%b%K%C%/(B, $BL5J?J}$G8_$$$KAG(B,

$B$?$@$7(B $c_i \in \C$ $B$OAj0[$J$j(B, $v_i \in \C[x]$, $v_i$ $B$O%b%K%C%/(B, $BL5J?J}$G8_$$$KAG(B,

$B$H=q$1$?$J$i$P(B, $c_i$ $B$O(B

$$R(z)=\res_x(n-zd',d) \in K[z]$$

$B$N:,$G(B,

Line 1306 $$v_i=\GCD(n-c_id',d).$$

Line 1305 $$v_i=\GCD(n-c_id',d).$$

$v=\prod_{i=1}^n$ $B$H$*$/$H(B,

$$nv = d\sum_{i=1}^nc_iv_i'(v/v_i).$$

$\GCD(n,d)=1$ $B$h$j(B $d|v.$ $B0lJ}$G(B, $v_i$|$B1&JU$h$j(B, $B$b$7(B $v{\not |}$

$\GCD(n,d)=1$ $B$h$j(B $d|v.$ $B0lJ}$G(B, $v_i|$$B1&JU$h$j(B, $B$b$7(B $v_i{\not |}d$

$B$J$i$P(B $v_i|c_iv_i'(v/v_i)$ $B$H$J$k$,(B, $B$3$l$O(B $v_i$ $B$K4X$9$k>r7o$h$jIT2DG=(B.

$B$h$C$F(B $v_i|d.$ $B7k6I(B $v|d$ $B$H$J$j(B $v=d.$ \qed\\

\underline{claim 2} $v_i=\GCD(n-c_id',d).$

Line 1328 $$g|(n-cd')=\sum_{j=1}^n (c_j-c)v_j'(v/v_j)$$

Line 1327 $$g|(n-cd')=\sum_{j=1}^n (c_j-c)v_j'(v/v_j)$$

$B$h$j(B, $g|(c_i-c)v_i'(v/v_i).$ $B$3$l$O(B $c_i=c$ $B$N$H$-$N$_2DG=(B. \qed

\begin{co}

$K$ $B$rJ#AG?tBN(B $C$ $B$NItJ,BN$H$7(B,

$K$ $B$rJ#AG?tBN(B $\C$ $B$NItJ,BN$H$7(B,

$f(x)=n(x)/d(x)$, $n,d \in K[x]$, $\GCD(n,d)=1$, $\deg(n)<\deg(d)$ $B$G(B $d$ $B$OL5J?J}(B, $B%b%K%C%/$H$7(B,

$$R(z)=\res_x(n-zd',d) \in K[z]$$

$B$H$9$k(B.