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數學中的對稱多項式是一种特殊的多元多项式。假设一个n元多項式P(X₁, X₂, ..., X_n)，當其中的n個不定元任意交換後，多項式仍維持不變，就称其为对称多项式。严格的说法是，如果对任意的n元置换σ，都有P(X_σ(1), X_σ(2), ..., X_σ(n)) = P(X₁, X₂, ..., X_n)，就说P是对称多项式。

对称多项式最早是在出现在对一元多项式方程求根的研究中。一元多项式方程的系数可以用它的根的多项式来表达。而多项式的任何一个根的地位理当与余者都相同，所以这类多项式中，不定元进行置换不应当改变多项式。从这个角度来说，将多项式方程的根构成的系数多项式称为基本对称多项式是合理的。有定理说明，任意的对称多项式都可以表达为基本对称多项式的多项式。

例子

• P(X1,X2)=X13+X23−7{displaystyle P(X_{1},X_{2})=X_{1}{}^{3}+X_{2}{}^{3}-7}


• P(X1,X2)=4X1X2{displaystyle P(X_{1},X_{2})=4X_{1}X_{2}}


• P(X1,X2,X3)=X1X2X3−2X1X2−2X1X3−2X2X3{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}}

以上的多項式都對稱。但是像P(X1,X2)=X1−2X2{displaystyle P(X_{1},X_{2})=X_{1}-2X_{2}}的多項式就不對稱，因為把X1{displaystyle X_{1}}和X2{displaystyle X_{2}}對換後，會得到X2−2X1{displaystyle X_{2}-2X_{1}}，不等於原來的多項式。

基本對稱多項式

對n{displaystyle n}個不定元X1,X2,...,Xn{displaystyle X_{1},X_{2},...,X_{n}}，有n{displaystyle n}個n{displaystyle n}元初等對稱多項式，就是(A+X1)(A+X2)...(A+Xn){displaystyle (A+X_{1})(A+X_{2})...(A+X_{n})}除首項外的各項係數。例如當n=3{displaystyle n=3}，基本對稱多項式為X1+X2+X3{displaystyle X_{1}+X_{2}+X_{3}}，X1X2+X2X3+X3X1{displaystyle X_{1}X_{2}+X_{2}X_{3}+X_{3}X_{1}}和X1X2X3{displaystyle X_{1}X_{2}X_{3}}。

基本對稱多項式是對稱多項式的構成單元。所有n{displaystyle n}元對稱多項式，都可以用這n{displaystyle n}個基本對稱多項式以加法和乘法表示出來。更準確地說：

任何n{displaystyle n}元對稱多項式，都可以用這n{displaystyle n}個以原來不定元組成的基本對稱多項式，唯一地以多項式來表示。

例如當n=2{displaystyle n=2}，有2個基本對稱多項式X1+X2{displaystyle X_{1}+X_{2}}和X1X2{displaystyle X_{1}X_{2}}。第一個例子中的多項式可以寫成

P(X1,X2)=X13+X23−7=(X1+X2)3−3X1X2(X1+X2)−7{displaystyle P(X_{1},X_{2})=X_{1}{}^{3}+X_{2}{}^{3}-7=(X_{1}+X_{2})^{3}-3X_{1}X_{2}(X_{1}+X_{2})-7}。

待定系数法

f(x1,x2,x3)=x13+x23+x33{displaystyle f(x_{1},x_{2},x_{3})=x_{1}^{3}+x_{2}^{3}+x_{3}^{3}}表达成基本对称多项式

设f(x1,x2,x3)=a(x1+x2+x3)3+b(x1+x2+x3)(x1x2+x1x3+x2x3)+c(x1x2x3){displaystyle f(x_{1},x_{2},x_{3})=a(x_{1}+x_{2}+x_{3})^{3}+b(x_{1}+x_{2}+x_{3})(x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})+c(x_{1}x_{2}x_{3})}

f(1,0,0)=a=1{displaystyle f(1,0,0)=a=1}

f(1,1,0)=8a+2b=2,b=−3{displaystyle f(1,1,0)=8a+2b=2,b=-3}

f(1,1,1)=27a+9b+c=3,c=3{displaystyle f(1,1,1)=27a+9b+c=3,c=3}^[1]

与高次方程的性质

F(x)=(x−x1)(x−x2)...(x−xn)=∑m=0namxm{displaystyle F(x)=(x-x_{1})(x-x_{2})...(x-x_{n})=sum _{m=0}^{n}a_{m}x^{m}}

F′(x)=(x−x2)...(x−xn)+(x−x1)(x−x3)...(x−xn)+...+(x−x1)(x−x2)...(x−xn−1){displaystyle 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−x3)(x−x4)...(x−xn)+(x−x2)(x−x4)...(x−xn)+...+(x−x1)(x−x2)...(x−xn−2){displaystyle 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})}

F′(x)F(x)=1x−x1+1x−x2+...+1x−xn{displaystyle {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

以下用a表示对称多项式，s表示等幂和：

∏r=1n(x−xr)=∑r=0narxr=0,sm=∑r=1nxrm{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

sm+a1sm−1+a2sm−2+...+am−1s1+mam=0{displaystyle s_{m}+a_{1}s_{m-1}+a_{2}s_{m-2}+...+a_{m-1}s_{1}+ma_{m}=0}^[2]

证明如下：

(∑i=1nkixir)∑i1≠i2≠...≠is−rxi1xi2...xis−r=∑i1≠i2≠...≠is−rki1xi1r+1xi2...xis−r+∑i1≠i2≠...≠is−rki1xi1rxi2...xis−r+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}}}

∑i1≠i2≠...≠is−1ki1xi12xi2...xis−1+∑i1≠i2≠...≠iski1xi11xi2...xis−∑i1≠i2≠...≠is−2ki1xi13xi2...xis−2−∑i1≠i2≠...≠is−1ki1xi12xi2...xis−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}}+...}

(−1)s−1∑i1ki1xi1s+∑i1≠i2≠...≠iski1xi11xi2...xis=∑r=1s−1(−1)r(∑i=1nkixir)∑i1≠i2≠...≠is−rxi1xi2...xis−r{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

两项时使等幂和分解为积与和的组合，如x12+x22=(x1+x2)2−2x1x2{displaystyle x_{1}^{2}+x_{2}^{2}=(x_{1}+x_{2})^{2}-2x_{1}x_{2}}：

x1m+x2m=∑r=0⌊m2⌋mCm−rrm−r(x1+x2)m−2r(−x1x2)r{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}}

用数学归纳法可证明高维的形式：

sm=∑ri=0⌊mi⌋m(r1+r2+...+rn−1)!r1!r2!...rn!∏i=1n(−an−i)ri{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}}}

f(m,r1,...,rn)=f(m−1,r1−1,...,rn)+...+f(m−n,r1,...,rn−1){displaystyle f(m,r_{1},...,r_{n})=f(m-1,r_{1}-1,...,r_{n})+...+f(m-n,r_{1},...,r_{n}-1)}

(m−1)(r1+...+rn−2)!(r1−1)!...rn!+...+(m−n)(r1+...+rn−2)!r1!...(rn−1)!=[r1(m−1)+...+rn(m−n)](r1+...+rn−2)!r1!...rn!{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}!}}}

=[m(r1+...+rn)−m](r1+...+rn−2)!r1!...rn!=m(r1+...+rn−1)!r1!...rn!{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}!}}}

取m=n=3{displaystyle m=n=3}

x13+x23+x33=3(3−1)!3!(x1+x2+x3)3+3(1+1−1)!1!1!(x1+x2+x3)(−x1x2−x1x3−x2x3)+3(1−1)!1!(x1x2x3){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})}

x13+x23+x33=(x1+x2+x3)3−3(x1+x2+x3)(x1x2+x1x3+x2x3)+3(x1x2x3){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})}

也可以把对称多项式表达成等幂和：

an−m=∑ri=0⌊mi⌋∏i=1m(−si)riiriri!{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}!}}}

mf(r1,...,rm)=f(r1−1,...,rm)+...+f(r1,...,rm−1){displaystyle mf(r_{1},...,r_{m})=f(r_{1}-1,...,r_{m})+...+f(r_{1},...,r_{m}-1)}

r1∏i=1m1iriri!+...+mrm∏i=1m1iriri!=m∏i=1m1iriri!{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}!}}=mprod _{i=1}^{m}{frac {1}{i^{r_{i}}r_{i}!}}}

取m=n=3{displaystyle m=n=3}

−x1x2x3=1133!(−x1−x2−x3)3+1111!211!(−x1−x2−x3)(−x12−x22−x32)+1311!(−x13−x23−x33){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})}

x1x2x3=16(x1+x2+x3)3−12(x1+x2+x3)(x12+x22+x32)+13(x13+x23+x33){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})}

参见

• 等幂求和


• 麦克劳林不等式

参考资料

查 论 编 基本乘法公式及恆等式 （因式分解） 分配律 (a+b)(c+d)=ac+ad+bc+bd{displaystyle (a+b)(c+d)=ac+ad+bc+bd,!} 二項式定理 和差平方 (a±b)2=a2±2ab+b2{displaystyle (apm b)^{2}=a^{2}pm 2ab+b^{2}} 和差立方 (a±b)3=a3±3a2b+3ab2±b3{displaystyle (apm b)^{3}=a^{3}pm 3a^{2}b+3ab^{2}pm b^{3}} 多項式定理 三數和平方 (a+b+c)2=a2+b2+c2+2ab+2bc+2ca{displaystyle (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ca,!} 等冪和差 平方和 a2+b2=(a+bi)(a−bi){displaystyle a^{2}+b^{2}=(a+bi)(a-bi)} 平方差 a2−b2=(a+b)(a−b){displaystyle a^{2}-b^{2}=left(a+bright)left(a-bright)} 立方和差 a3±b3=(a±b)(a2∓ab+b2){displaystyle a^{3}pm b^{3}=(apm b)(a^{2}mp ab+b^{2}),!} 等冪和差逆定理 a4+a2b2+b4=(a2+ab+b2)(a2−ab+b2){displaystyle a^{4}+a^{2}b^{2}+b^{4}=(a^{2}+ab+b^{2})(a^{2}-ab+b^{2}),!} 對稱多項式 a3+b3+c3=(a+b+c)3+3(a+b+c)(−ab−bc−ca)+3abc{displaystyle a^{3}+b^{3}+c^{3}=(a+b+c)^{3}+3(a+b+c)(-ab-bc-ca)+3abc,!}