對稱多項式

例子

• $P(X_1, X_2) = X_1{}^3+ X_2{}^3-7$
• $P(X_1, X_2) = 4 X_1 X_2$
• $P(X_1, X_2, X_3) = X_1 X_2 X_3 - 2 X_1 X_2 - 2 X_1 X_3 - 2 X_2 X_3$

基本對稱多項式

$n$個不定元$X_1, X_2, ..., X_n$，有$n$$n$初等對稱多項式，就是$(A+X_1)(A+X_2)...(A+X_n)$除首項外的各項係數。例如當$n=3$，基本對稱多項式為$X_1+X_2+X_3$$X_1X_2 + X_2X_3 + X_3X_1$$X_1X_2X_3$

$P(X_1, X_2) = X_1{}^3+ X_2{}^3-7=(X_1+X_2)^3-3X_1X_2(X_1+X_2)-7$

与高次方程的性质

$F(x)= (x-x_1)(x-x_2)...(x-x_n)=\sum_{m=0}^{n}a_mx^m$

$F'(x)=(x-x_2)...(x-x_n)+(x-x_1)(x-x_3)...(x-x_n)+...+(x-x_1)(x-x_2)...(x-x_{n-1})$

$F''(x)=(x-x_3)(x-x_4)...(x-x_n)+(x-x_2)(x-x_4)...(x-x_n)+...+(x-x_1)(x-x_2)...(x-x_{n-2})$

$\frac{F'(x)}{F(x)}=\frac{1}{x-x_1}+\frac{1}{x-x_2}+...+\frac{1}{x-x_n}$

与等幂和的性质

$\prod_{r=1}^n (x-x_r)=\sum_{r=0}^n a_r x^r=0,s_m=\sum_{r=1}^n x_r^m$

牛顿公式

$s_m+a_1s_{m-1}+a_2s_{m-2}+...+a_{m-1}s_1+ma_m=0$[1]

$\displaystyle(\sum_{i=1}^n k_i x_i^r)\sum_{i_1 \neq i_2 \neq ... \neq i_{s-r}} x_{i_1}x_{i_2}...x_{i_{s-r}} =\sum_{i_1 \neq i_2 \neq ... \neq i_{s-r}} k_{i_1}x_{i_1}^{r+1}x_{i_2}...x_{i_{s-r}} +\sum_{i_1 \neq i_2 \neq ... \neq i_{s-r}} k_{i_1}x_{i_1}^{r}x_{i_2}...x_{i_{s-r+1}}$

$\displaystyle\sum_{i_1 \neq i_2 \neq ... \neq i_{s-1}} k_{i_1}x_{i_1}^{2}x_{i_2}...x_{i_{s-1}}+\sum_{i_1 \neq i_2 \neq ... \neq i_{s}} k_{i_1}x_{i_1}^{1}x_{i_2}...x_{i_{s}}-\sum_{i_1 \neq i_2 \neq ... \neq i_{s-2}} k_{i_1}x_{i_1}^{3}x_{i_2}...x_{i_{s-2}}-\sum_{i_1 \neq i_2 \neq ... \neq i_{s-1}} k_{i_1}x_{i_1}^{2}x_{i_2}...x_{i_{s-1}}+...$

$\displaystyle(-1)^{s-1}\sum_{i_1} k_{i_1}x_{i_1}^{s}+\sum_{i_1 \neq i_2 \neq ... \neq i_{s}} k_{i_1}x_{i_1}^{1}x_{i_2}...x_{i_{s}}=\sum_{r=1}^{s-1} (-1)^r (\sum_{i=1}^n k_i x_i^r)\sum_{i_1 \neq i_2 \neq ... \neq i_{s-r}} x_{i_1}x_{i_2}...x_{i_{s-r}}$

组合公式

$x_1^m+x_2^m=\sum_{r=0}^{\lfloor \frac{m}{2} \rfloor}\frac{mC_{m-r}^{r}}{m-r}(x_1+x_2)^{m-2r}(-x_1 x_2)^r$

$s_m=\sum_{r_i=0}^{\lfloor \frac{m}{i} \rfloor} \frac{m(r_1+r_2+...+r_n -1)!}{r_1!r_2!...r_n!} \prod_{i=1}^n (-a_{n-i})^{r_i}$
$f(m,r_1,...,r_n)=f(m-1,r_1-1,...,r_n)+...+f(m-n,r_1,...,r_n-1)$
$\frac{(m-1)(r_1+...+r_n-2)!}{(r_1-1)!...r_n!}+...+\frac{(m-n)(r_1+...+r_n-2)!}{r_1!...(r_n-1)!}=\frac{[r_1(m-1)+...+r_n(m-n)](r_1+...+r_n-2)!}{r_1!...r_n!}$
$=\frac{[m(r_1+...+r_n)-m](r_1+...+r_n-2)!}{r_1!...r_n!}=\frac{m(r_1+...+r_n-1)!}{r_1!...r_n!}$

$a_{n-m}=\sum_{r_i=0}^{\lfloor \frac{m}{i} \rfloor} \prod_{i=1}^m \frac{(-s_i)^{r_i}}{i^{r_i} r_i !}$
$mf(r_1,...,r_m)=f(r_1-1,...,r_m)+...+f(r_1,...,r_m-1)$
$r_1\prod_{i=1}^m \frac{1}{i^{r_i} r_i!}+...+mr_m\prod_{i=1}^m \frac{1}{i^{r_i} r_i!}=m\prod_{i=1}^m \frac{1}{i^{r_i} r_i!}$