# 对称多项式

## 例子

• ${\displaystyle P(X_{1},X_{2})=X_{1}{}^{3}+X_{2}{}^{3}-7}$
• ${\displaystyle P(X_{1},X_{2})=4X_{1}X_{2))$

• ${\displaystyle P(X_{1},X_{2},X_{3})=X_{1}X_{2}X_{3}-2X_{1}X_{2}-2X_{1}X_{3}-2X_{2}X_{3))$

${\displaystyle D(X_{1},\dots ,X_{n})=\prod _{1\leq i

## 单变数首一多项式的根

${\displaystyle f(t)=t^{n}+a_{n-1}t^{n-1}+\cdots +a_{2}t^{2}+a_{1}t+a_{0))$

${\displaystyle t^{n}+a_{n-1}t^{n-1}+\cdots +a_{2}t^{2}+a_{1}t+a_{0}=(t-x_{1})(t-x_{2})\cdots (t-x_{n})}$

{\displaystyle {\begin{aligned}a_{n-1}&=-x_{1}-x_{2}-\cdots -x_{n}\\a_{n-2}&=x_{1}x_{2}+x_{1}x_{3}+\cdots +x_{2}x_{3}+\cdots +x_{n-1}x_{n}=\textstyle \sum _{1\leq i

${\displaystyle (t+X_{1})(t+X_{2})\cdots (t+X_{n})}$

## 对称多项式基本定理

${\displaystyle X_{1}^{3}+X_{2}^{3}-7=(X_{1}+X_{2})^{3}-3X_{1}X_{2}(X_{1}+X_{2})-7}$

## 一些常见的对称多项式

### 单项对称多项式

${\displaystyle m_{(3,1,1)}(X_{1},X_{2},X_{3})=X_{1}^{3}X_{2}X_{3}+X_{1}X_{2}^{3}X_{3}+X_{1}X_{2}X_{3}^{3))$,
${\displaystyle m_{(3,2,1)}(X_{1},X_{2},X_{3})=X_{1}^{3}X_{2}^{2}X_{3}+X_{1}^{3}X_{2}X_{3}^{2}+X_{1}^{2}X_{2}^{3}X_{3}+X_{1}^{2}X_{2}X_{3}^{3}+X_{1}X_{2}^{3}X_{3}^{2}+X_{1}X_{2}^{2}X_{3}^{3}.}$

${\displaystyle e_{k}(X_{1},\ldots ,X_{n})=m_{\alpha }(X_{1},\ldots ,X_{n})}$

### 次方和对称多项式

${\displaystyle p_{k}(X_{1},\ldots ,X_{n})=X_{1}^{k}+X_{2}^{k}+\cdots +X_{n}^{k}.}$

${\displaystyle p_{3}(X_{1},X_{2})=\textstyle {\frac {3}{2))p_{2}(X_{1},X_{2})p_{1}(X_{1},X_{2})-{\frac {1}{2))p_{1}(X_{1},X_{2})^{3}.}$

${\displaystyle m_{(2,1)}(X_{1},X_{2})=X_{1}^{2}X_{2}+X_{1}X_{2}^{2))$

${\displaystyle m_{(2,1)}(X_{1},X_{2})=\textstyle {\frac {1}{2))p_{1}(X_{1},X_{2})^{3}-{\frac {1}{2))p_{2}(X_{1},X_{2})p_{1}(X_{1},X_{2}).}$

{\displaystyle {\begin{aligned}m_{(2,1)}(X_{1},X_{2},X_{3})&=X_{1}^{2}X_{2}+X_{1}X_{2}^{2}+X_{1}^{2}X_{3}+X_{1}X_{3}^{2}+X_{2}^{2}X_{3}+X_{2}X_{3}^{2}\\&=p_{1}(X_{1},X_{2},X_{3})p_{2}(X_{1},X_{2},X_{3})-p_{3}(X_{1},X_{2},X_{3}).\end{aligned))}

## 初等对称多项式

### 与等幂和的性质

${\displaystyle \prod _{r=1}^{n}(x-x_{r})=\sum _{r=0}^{n}a_{r}x^{r}=0,s_{m}=\sum _{r=1}^{n}x_{r}^{m))$

#### 牛顿公式

${\displaystyle s_{m}+a_{1}s_{m-1}+a_{2}s_{m-2}+...+a_{m-1}s_{1}+ma_{m}=0}$[1]

${\displaystyle \displaystyle (\sum _{i=1}^{n}k_{i}x_{i}^{r})\sum _{i_{1}\neq i_{2}\neq ...\neq i_{s-r))x_{i_{1))x_{i_{2))...x_{i_{s-r))=\sum _{i_{1}\neq i_{2}\neq ...\neq i_{s-r))k_{i_{1))x_{i_{1))^{r+1}x_{i_{2))...x_{i_{s-r))+\sum _{i_{1}\neq i_{2}\neq ...\neq i_{s-r))k_{i_{1))x_{i_{1))^{r}x_{i_{2))...x_{i_{s-r+1))}$

${\displaystyle \displaystyle \sum _{i_{1}\neq i_{2}\neq ...\neq i_{s-1))k_{i_{1))x_{i_{1))^{2}x_{i_{2))...x_{i_{s-1))+\sum _{i_{1}\neq i_{2}\neq ...\neq i_{s))k_{i_{1))x_{i_{1))^{1}x_{i_{2))...x_{i_{s))-\sum _{i_{1}\neq i_{2}\neq ...\neq i_{s-2))k_{i_{1))x_{i_{1))^{3}x_{i_{2))...x_{i_{s-2))-\sum _{i_{1}\neq i_{2}\neq ...\neq i_{s-1))k_{i_{1))x_{i_{1))^{2}x_{i_{2))...x_{i_{s-1))+...}$

${\displaystyle \displaystyle (-1)^{s-1}\sum _{i_{1))k_{i_{1))x_{i_{1))^{s}+\sum _{i_{1}\neq i_{2}\neq ...\neq i_{s))k_{i_{1))x_{i_{1))^{1}x_{i_{2))...x_{i_{s))=\sum _{r=1}^{s-1}(-1)^{r}(\sum _{i=1}^{n}k_{i}x_{i}^{r})\sum _{i_{1}\neq i_{2}\neq ...\neq i_{s-r))x_{i_{1))x_{i_{2))...x_{i_{s-r))}$

#### 组合公式

${\displaystyle x_{1}^{m}+x_{2}^{m}=\sum _{r=0}^{\lfloor {\frac {m}{2))\rfloor }{\frac {mC_{m-r}^{r)){m-r))(x_{1}+x_{2})^{m-2r}(-x_{1}x_{2})^{r))$

${\displaystyle s_{m}=\sum _{r_{i}=0}^{\lfloor {\frac {m}{i))\rfloor }{\frac {m(r_{1}+r_{2}+...+r_{n}-1)!}{r_{1}!r_{2}!...r_{n}!))\prod _{i=1}^{n}(-a_{n-i})^{r_{i))}$
${\displaystyle f(m,r_{1},...,r_{n})=f(m-1,r_{1}-1,...,r_{n})+...+f(m-n,r_{1},...,r_{n}-1)}$
${\displaystyle {\frac {(m-1)(r_{1}+...+r_{n}-2)!}{(r_{1}-1)!...r_{n}!))+...+{\frac {(m-n)(r_{1}+...+r_{n}-2)!}{r_{1}!...(r_{n}-1)!))={\frac {[r_{1}(m-1)+...+r_{n}(m-n)](r_{1}+...+r_{n}-2)!}{r_{1}!...r_{n}!))}$
${\displaystyle ={\frac {[m(r_{1}+...+r_{n})-m](r_{1}+...+r_{n}-2)!}{r_{1}!...r_{n}!))={\frac {m(r_{1}+...+r_{n}-1)!}{r_{1}!...r_{n}!))}$

${\displaystyle m=n=3}$

${\displaystyle \displaystyle x_{1}^{3}+x_{2}^{3}+x_{3}^{3}={\frac {3(3-1)!}{3!))(x_{1}+x_{2}+x_{3})^{3}+{\frac {3(1+1-1)!}{1!1!))(x_{1}+x_{2}+x_{3})(-x_{1}x_{2}-x_{1}x_{3}-x_{2}x_{3})+{\frac {3(1-1)!}{1!))(x_{1}x_{2}x_{3})}$
${\displaystyle x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=(x_{1}+x_{2}+x_{3})^{3}-3(x_{1}+x_{2}+x_{3})(x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})+3(x_{1}x_{2}x_{3})}$

${\displaystyle a_{n-m}=\sum _{r_{i}=0}^{\lfloor {\frac {m}{i))\rfloor }\prod _{i=1}^{m}{\frac {(-s_{i})^{r_{i))}{i^{r_{i))r_{i}!))}$
${\displaystyle mf(r_{1},...,r_{m})=f(r_{1}-1,...,r_{m})+...+f(r_{1},...,r_{m}-1)}$
${\displaystyle r_{1}\prod _{i=1}^{m}{\frac {1}{i^{r_{i))r_{i}!))+...+mr_{m}\prod _{i=1}^{m}{\frac {1}{i^{r_{i))r_{i}!))=m\prod _{i=1}^{m}{\frac {1}{i^{r_{i))r_{i}!))}$

${\displaystyle m=n=3}$

${\displaystyle \displaystyle -x_{1}x_{2}x_{3}={\frac {1}{1^{3}3!))(-x_{1}-x_{2}-x_{3})^{3}+{\frac {1}{1^{1}1!2^{1}1!))(-x_{1}-x_{2}-x_{3})(-x_{1}^{2}-x_{2}^{2}-x_{3}^{2})+{\frac {1}{3^{1}1!))(-x_{1}^{3}-x_{2}^{3}-x_{3}^{3})}$
${\displaystyle \displaystyle x_{1}x_{2}x_{3}={\frac {1}{6))(x_{1}+x_{2}+x_{3})^{3}-{\frac {1}{2))(x_{1}+x_{2}+x_{3})(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})+{\frac {1}{3))(x_{1}^{3}+x_{2}^{3}+x_{3}^{3})}$

## 参考资料

1. ^ 沈南山. 牛顿(Newton)公式的一个注记及其应用. 数学通报. 2005, (3).