対称式の基本対称式による表示

代数学序論（東海大学情報数理学科, 担当：那須）

対称式を基本対称式で表す命令

SymmetricReduction[f,{(Subscript[x, 1]),…,(Subscript[x, n])},{(Subscript[s, 1]),…,(Subscript[s, n])}]
変数(Subscript[x, 1]),…,(Subscript[x, n])に関する対称式p を基本対称式 (Subscript[s, 1]),…,(Subscript[s, n])で置き換えたペア{p,q}を与える

2変数(Binomial symmmetric functions)

2次式

SymmetricReduction[x^2+y^2,{x,y},{s1,s2}]

3次式

SymmetricReduction[x^3+y^3,{x,y},{s1,s2}]

4次式

SymmetricReduction[x^4+y^4,{x,y},{s1,s2}]

5次式

SymmetricReduction[x^5+y^5,{x,y},{s1,s2}]

3変数

2次式

SymmetricReduction[x^2+y^2+z^2,{x,y,z},{s1,s2,s3}]

SymmetricReduction[(x-y)^2+(y-z)^2+(z-x)^2,{x,y,z},{s1,s2,s3}]

SymmetricReduction[(x+y)^2+(y+z)^2+(z+x)^2,{x,y,z},{s1,s2,s3}]

SymmetricReduction[Simplify[Det[{{x,y,z},{x^3,y^3,z^3},{x^4,y^4,z^4}}]/Det[{{x,y,z},{x^2,y^2,z^2},{x^3,y^3,z^3}}]],{x,y,z},{s1,s2,s3}]

SymmetricReduction[Simplify[Det[{{x,y,z},{x^2,y^2,z^2},{x^5,y^5,z^5}}]/Det[{{x,y,z},{x^2,y^2,z^2},{x^3,y^3,z^3}}]],{x,y,z},{s1,s2,s3}]

3次式

SymmetricReduction[x^3+y^3+z^3,{x,y,z},{s1,s2,s3}]

SymmetricReduction[z(x-y)^2+x(y-z)^2+y(z-x)^2,{x,y,z},{s1,s2,s3}]

SymmetricReduction[z(x+y)^2+x(y+z)^2+y(z+x)^2,{x,y,z},{s1,s2,s3}]

SymmetricReduction[z(x^2+y^2)+x(y^2+z^2)+y(z^2+x^2),{x,y,z},{s1,s2,s3}]

SymmetricReduction[z^2(x+y)+x^2(y+z)+y^2(z+x),{x,y,z},{s1,s2,s3}]

SymmetricReduction[(x+y)^3+(y+z)^3+(z+x)^3,{x,y,z},{s1,s2,s3}]

SymmetricReduction[(x+y)(y+z)(z+x)+x*y*z,{x,y,z},{s1,s2,s3}]

SymmetricReduction[Det[{{x,y,z},{z,x,y},{y,z,x}}],{x,y,z},{s1,s2,s3}]

SymmetricReduction[Simplify[Det[{{x^2,y^2,z^2},{x^3,y^3,z^3},{x^4,y^4,z^4}}]/Det[{{x,y,z},{x^2,y^2,z^2},{x^3,y^3,z^3}}]],{x,y,z},{s1,s2,s3}]

4次式

SymmetricReduction[x^4+y^4+z^4,{x,y,z},{s1,s2,s3}]

SymmetricReduction[x^2y^2+y^2z^2+z^2x^2,{x,y,z},{s1,s2,s3}]

SymmetricReduction[(x-y)^4+(y-z)^4+(z-x)^4,{x,y,z},{s1,s2,s3}]

SymmetricReduction[(x+y)^4+(y+z)^4+(z+x)^4,{x,y,z},{s1,s2,s3}]

SymmetricReduction[x^2(y-z)^2+y^2*(z-x)^2+z^2*(x-y)^2,{x,y,z},{s1,s2,s3}]

SymmetricReduction[x^2(y+z)^2+y^2*(z+x)^2+z^2*(x+y)^2,{x,y,z},{s1,s2,s3}]

SymmetricReduction[x(y+z)^3+y*(z+x)^3+z*(x+y)^3,{x,y,z},{s1,s2,s3}]

SymmetricReduction[x^3(y+z)+y^3*(z+x)+z^3*(x+y),{x,y,z},{s1,s2,s3}]

SymmetricReduction[x*y*(x-y)^2+y*z*(y-z)^2+z*x*(z-x)^2,{x,y,z},{s1,s2,s3}]

SymmetricReduction[Simplify[Det[{{x,y,z},{x^4,y^4,z^4},{x^5,y^5,z^5}}]/Det[{{x,y,z},{x^2,y^2,z^2},{x^3,y^3,z^3}}]],{x,y,z},{s1,s2,s3}]

5次式

SymmetricReduction[x^5+y^5+z^5,{x,y,z},{s1,s2,s3}]

4変数

2次式

SymmetricReduction[x^2+y^2+z^2+w^2,{x,y,z,w},{s1,s2,s3,s4}]

SymmetricReduction[(x-y)^2+(y-z)^2+(z-x)^2+(x-w)^2+(y-w)^2+(z-w)^2,{x,y,z,w},{s1,s2,s3,s4}]

SymmetricReduction[(x+y)^2+(y+z)^2+(z+x)^2+(x+w)^2+(y+w)^2+(z+w)^2,{x,y,z,w},{s1,s2,s3,s4}]

3次式

SymmetricReduction[x^3+y^3+z^3+w^3,{x,y,z,w},{s1,s2,s3,s4}]

SymmetricReduction[(x+y-z-w)(x-y+z-w)(x-y-z+w),{x,y,z,w},{s1,s2,s3,s4}]