table of Contents
definition
Mathematics is a special symmetric polynomials multivariate polynomial.
If a polynomial n \ (\ text P (x_1, x_2, ..., x_n) \) , for arbitrary permutations n \ (\ Sigma \) , are \ (\ text P (x _ {\ sigma_1} , X _ {\ sigma_2}, ..., {X _ \ sigma_n}) = \ text P (x_1, x_2, ..., x_n) \) , said \ (\ text P \) is a symmetric polynomial.
Some (special) example
Van der Monde matrix determinant: \ (\ Prod \ {limits_. 1 \ Le I <J \} n-Le (x_j-x_i) \) , this thing is incidentally polynomial \ (\ prod \ limits_ {i = 1 } ^ n (x-x_i) \) discriminant
idempotent symmetrical formula: \ (P_K (x_1, x_2, ..., x_n) = \ SUM \ limits_ = {I}. 1 nx_i ^ ^ K \)
elementary symmetric formula: \ (E_k (x_1, x_2, ..., x_n) = \ SUM \ limits_ {S \ subseteq \ {1,2, ..., n-\}, | S | = K} \ Prod \ {limits_ I \ S} in x_i,. 1 \ Le K \ n-Le \) , when the \ (k> n \) when \ (E_k (x_1, x_2, ..., x_n) = 0 \) , when the \ (k = 0 \) when, \ (. 1 E_k = \)
is completely homogeneous symmetrical formula: \ (h_k (x_1, x_2, ..., x_n) = \ SUM \ {limits_. 1 \ Le i_1 \ Le i_2 are used \ Le \ cdots \ i_k Le \ Le n-X_ {i_1}}} ... X_ {i_2 are used i_k X_ {} \) ,, when the \ (k = 0 \) when, \ (= h_k. 1 \)
Fundamental theorem of symmetric polynomials
N F is a polynomial algebraic number n is symmetrical Yuanchu like, if and only if F is a symmetrical polynomial
immediate consequence of this theorem: a first n-th root of a symmetric polynomial into a polynomial, each equal to the original polynomial into a polynomial coefficient
Some interesting conclusions
Since the n-th argument discussed are the same, hereinafter, the idempotent symmetrical, elementary symmetric, full elementary symmetric section \ (K \) which respectively abbreviated as \ (p_k, e_k, h_k \
) Order \ (P ( x) = \ sum \ limits_ { i = 0} ^ {+ \ infty} p_ix ^ i, E_0 (x) = \ sum \ limits_ {i = 0} ^ {+ \ infty} e_ix ^ i, H (x) = \ sum \ limits_ {i = 0} ^ {+ \ infty} h_ix ^ i, E (x) = E_0 (-x) \)
Conclusion 1
\ (E * H = 1 \
) Proof:
apparently \ (H (x) = \ prod \ limits_ {i = 1} ^ {n} \ frac1 {1-x_ix} \) and \ (F (x) = \ prod \ limits_ {i = 1
} ^ n (x-x_i) = \ sum \ limits_ {i = 0} ^ {+ \ infty} (- 1) ^ ie_ix ^ {ni} \) then \ (x ^ nF (\ frac1x) = \ prod \ limits_ {i = 1} ^ n (1-x_ix) = \ sum \ limits_ {i = 0} ^ {+ \ infty} (- 1) ^ ie_ix ^ i = E (x) \)
then \ (E * H = 1 \
) is proved
Conclusion 2
\ [\ Begin {aligned} e_k (1,2, ..., n) = & S1_ {n + 1} ^ {n + 1-k} \\ h_k (1,2, ..., n) = & S2_ {n + k} ^ {n
} \ end {aligned} \] the ratio of the generated function can be demonstrated
Conclusion 3
\ (\ forall n, k \
ge1, ke_k = \ sum \ limits_ {i = 1} ^ k (-1) ^ {i-1} e_ {ki} p_i \) can be proved by mathematical induction, from a unified push the n-ary form
this conclusion may be in the form of further generating function deduced \ (P (the -X-) = (\ E_0 LN (X)) '\) , which can quickly achieve \ (P, E_0 \) intermetallic