Multivariable generalization of Newton's identities
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Given a finite sequence, ${a_i ;|; i=1,...,N}$, Newton's identities relate the power sums, $p_k = sum_{i=1}^N {a_i}^k$, and the elementary symmetric polynomials, $e_k = sum_{i_1 < ... < i_k} a_{i_1} ... a_{i_k}$. Is there a generalization to the multivariable case? Say, if I have two sequences, ${a_i ;|; k=1,...,N}$ and ${b_i ;|; i=1,...,N}$, and given the power sums (say, only for $k,ell>0$, otherwise this reduces to the single variable case):
$$ p_{k,ell} = sum_{i=1}^N {a_i}^k {b_i}^ell, $$
is there any simple way to extract the elementary symmetric polynomials in the $a_i$ and $b_i$?
abstract-algebra
asked Nov 17 at 17:21
user6013
1583
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up vote
0
down vote
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Given a finite sequence, ${a_i ;|; i=1,...,N}$, Newton's identities relate the power sums, $p_k = sum_{i=1}^N {a_i}^k$, and the elementary symmetric polynomials, $e_k = sum_{i_1 < ... < i_k} a_{i_1} ... a_{i_k}$. Is there a generalization to the multivariable case? Say, if I have two sequences, ${a_i ;|; k=1,...,N}$ and ${b_i ;|; i=1,...,N}$, and given the power sums (say, only for $k,ell>0$, otherwise this reduces to the single variable case):
$$ p_{k,ell} = sum_{i=1}^N {a_i}^k {b_i}^ell, $$
is there any simple way to extract the elementary symmetric polynomials in the $a_i$ and $b_i$?
abstract-algebra
asked Nov 17 at 17:21
user6013
1583
mathoverflow.net/questions/304727/… gives a formula for the elementaries in terms of the power sums. Simple? You decide.
– darij grinberg
Nov 17 at 17:25
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given a finite sequence, ${a_i ;|; i=1,...,N}$, Newton's identities relate the power sums, $p_k = sum_{i=1}^N {a_i}^k$, and the elementary symmetric polynomials, $e_k = sum_{i_1 < ... < i_k} a_{i_1} ... a_{i_k}$. Is there a generalization to the multivariable case? Say, if I have two sequences, ${a_i ;|; k=1,...,N}$ and ${b_i ;|; i=1,...,N}$, and given the power sums (say, only for $k,ell>0$, otherwise this reduces to the single variable case):
$$ p_{k,ell} = sum_{i=1}^N {a_i}^k {b_i}^ell, $$
is there any simple way to extract the elementary symmetric polynomials in the $a_i$ and $b_i$?
abstract-algebra
asked Nov 17 at 17:21
user6013
1583
Given a finite sequence, ${a_i ;|; i=1,...,N}$, Newton's identities relate the power sums, $p_k = sum_{i=1}^N {a_i}^k$, and the elementary symmetric polynomials, $e_k = sum_{i_1 < ... < i_k} a_{i_1} ... a_{i_k}$. Is there a generalization to the multivariable case? Say, if I have two sequences, ${a_i ;|; k=1,...,N}$ and ${b_i ;|; i=1,...,N}$, and given the power sums (say, only for $k,ell>0$, otherwise this reduces to the single variable case):
$$ p_{k,ell} = sum_{i=1}^N {a_i}^k {b_i}^ell, $$
is there any simple way to extract the elementary symmetric polynomials in the $a_i$ and $b_i$?
abstract-algebra
abstract-algebra
asked Nov 17 at 17:21
user6013
1583
asked Nov 17 at 17:21
user6013
1583
asked Nov 17 at 17:21
user6013
1583
asked Nov 17 at 17:21
user6013
1583
asked Nov 17 at 17:21
user6013
1583
1583
mathoverflow.net/questions/304727/… gives a formula for the elementaries in terms of the power sums. Simple? You decide.
– darij grinberg
Nov 17 at 17:25
add a comment |
mathoverflow.net/questions/304727/… gives a formula for the elementaries in terms of the power sums. Simple? You decide.
– darij grinberg
Nov 17 at 17:25
mathoverflow.net/questions/304727/… gives a formula for the elementaries in terms of the power sums. Simple? You decide.
– darij grinberg
Nov 17 at 17:25
mathoverflow.net/questions/304727/… gives a formula for the elementaries in terms of the power sums. Simple? You decide.
– darij grinberg
Nov 17 at 17:25
add a comment |
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mathoverflow.net/questions/304727/… gives a formula for the elementaries in terms of the power sums. Simple? You decide.
– darij grinberg
Nov 17 at 17:25