完全对称多项式、初等对称多项式的关系

hbghlyj

完全对称多项式定义为

1. completeSymmetricPolynomial[i_?IntegerQ, vars_?ListQ] :=
2.       Total@Union@Tuples[Times @@ vars, {i}];

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例如

1. completeSymmetricPolynomial[2, {a, b, c, d}]

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$$=a^2 + a b + b^2 + a c + b c + c^2 + a d + b d + c d + d^2$$

hbghlyj

设$e_i$为初等对称多项式，如何证明$\displaystyle\sum_{i=0}^m (-1)^i e_i(X_1,\dots,X_n)h_{m-i}(X_1,\dots,X_n)=0$
例如

1. FullSimplify@
2. Table[Sum[(-1)^i completeSymmetricPolynomial[i, {a, b, c, d}]     
3.                           SymmetricPolynomial[m - i, {a, b, c, d}],
4.       {i, 0, m}], {m, 1, 4}]

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{0, 0, 0, 0}