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| In [[mathematics]], '''Newton's identities''', also known as the '''Newton–Girard formulae''', give relations between two types of [[symmetric polynomial]]s, namely between [[Power sum symmetric polynomial|power sums]] and [[elementary symmetric polynomial]]s. Evaluated at the [[Root of a function|root]]s of a monic [[polynomial]] ''P'' in one variable, they allow expressing the sums of the ''k''-th [[power (mathematics)|power]]s of all roots of ''P'' (counted with their multiplicity) in terms of the coefficients of ''P'', without actually finding those roots. These identities were found by [[Isaac Newton]] around 1666, apparently in ignorance of earlier work (1629) by [[Albert Girard]]. They have applications in many areas of mathematics, including [[Galois theory]], [[invariant theory]], [[group theory]], [[combinatorics]], as well as further applications outside mathematics, including [[general relativity]].
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| == Mathematical statement ==
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| === Formulation in terms of symmetric polynomials ===
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| Let ''x''<sub>1</sub>,…, ''x''<sub>''n''</sub> be variables, denote for ''k'' ≥ 1 by ''p''<sub>''k''</sub>(''x''<sub>1</sub>,…,''x''<sub>''n''</sub>) the ''k''-th '''power sum''':
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| : <math>p_k(x_1,\ldots,x_n)=\sum\nolimits_{i=1}^nx_i^k = x_1^k+\cdots+x_n^k,</math>
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| and for ''k'' ≥ 0 denote by ''e''<sub>''k''</sub>(''x''<sub>1</sub>,…,''x''<sub>''n''</sub>) the [[elementary symmetric polynomial]] that is the sum of all distinct products of ''k'' distinct variables, so in particular
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| : <math>\begin{align}
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| e_0(x_1,\ldots,x_n) &= 1,\\
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| e_1(x_1,\ldots,x_n) &= x_1+x_2+\cdots+x_n,\\
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| e_2(x_1,\ldots,x_n) &= \textstyle\sum_{1 \leq i<j\leq n}x_ix_j,\\
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| e_n(x_1,\ldots,x_n) &= x_1x_2\cdots x_n,\\
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| e_k(x_1,\ldots,x_n) &= 0, \quad\text{for}\ k>n.\\
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| \end{align}</math>
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| Then Newton's identities can be stated as
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| : <math> ke_k(x_1,\ldots,x_n) = \sum_{i=1}^k(-1)^{i-1} e_{k-i} (x_1,\ldots,x_n) p_i(x_1,\ldots,x_n),</math>
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| valid for all ''k'' ≥ 1. Concretely, one gets for the first few values of ''k'':
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| : <math>\begin{align}
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| e_1(x_1,\ldots,x_n) &= p_1(x_1,\ldots,x_n),\\
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| 2e_2(x_1,\ldots,x_n) &= e_1(x_1,\ldots,x_n)p_1(x_1,\ldots,x_n)-p_2(x_1,\ldots,x_n),\\
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| 3e_3(x_1,\ldots,x_n) &= e_2(x_1,\ldots,x_n)p_1(x_1,\ldots,x_n) - e_1(x_1,\ldots,x_n)p_2(x_1,\ldots,x_n) + p_3(x_1,\ldots,x_n).\\
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| \end{align}</math>
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| The form and validity of these equations do not depend on the number ''n'' of variables (although the point where the left-hand side becomes 0 does, namely after the ''n''-th identity), which makes it possible to state them as identities in the [[ring of symmetric functions]]. In that ring one has
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| :<math>\begin{align}
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| e_1 &= p_1,\\
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| 2e_2 &= e_1p_1-p_2,\\ | |
| 3e_3 &= e_2p_1 - e_1p_2 + p_3,\\
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| 4e_4 &= e_3p_1 - e_2p_2 + e_1p_3 - p_4,\\
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| \end{align}</math>
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| and so on; here the left-hand sides never become zero.
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| These equations allow to recursively express the ''e''<sub>''i''</sub> in terms of the ''p''<sub>''k''</sub>; to be able to do the inverse, one may rewrite them as
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| :<math>\begin{align}
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| p_1 &= e_1,\\
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| p_2 &= e_1p_1-2e_2,\\
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| p_3 &= e_1p_2 - e_2p_1 + 3e_3 ,\\
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| p_4 &= e_1p_3 - e_2p_2 + e_3p_1 - 4e_4, \\
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| & {}\ \ \vdots
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| \end{align}</math>
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| === Application to the roots of a polynomial ===
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| Now view the ''x''<sub>''i''</sub> as parameters rather than as variables, and consider the monic polynomial in ''t'' with roots ''x''<sub>1</sub>,…,''x''<sub>''n''</sub>:
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| :<math> \prod_{i=1}^n \left( t - x_i \right) = \sum_{k=0}^n (-1)^{k} a_k t^{n-k},</math>
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| where the [[coefficients]] <math>a_k</math> are given by the elementary symmetric polynomials in the roots: ''a''<sub>''k''</sub> = ''e''<sub>''k''</sub>(''x''<sub>1</sub>,…,''x''<sub>''n''</sub>).
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| Now consider the ''power sums'' of the roots
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| :<math> s_k = p_k(x_1,\ldots,x_n)= \sum_{i=1}^n x_i^k.</math>
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| Then according to Newton's identities these can be expressed recursively in terms of the coefficients of the polynomial using
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| :<math>\begin{align}
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| s_1 &= a_1,\\
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| s_2 &= a_1 s_1 - 2 a_2,\\
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| s_3 &= a_1 s_2 - a_2 s_1 + 3 a_3,\\
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| s_4 &= a_1 s_3 - a_2 s_2 + a_3 s_1 - 4 a_4, \\
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| & {} \ \ \vdots
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| \end{align}</math>
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| === Application to the characteristic polynomial of a matrix ===
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| When the polynomial above is the [[characteristic polynomial]] of a [[matrix (mathematics)|matrix]] ''A'', the roots <math>x_i</math> are the [[eigenvalue]]s of the matrix, counted with their algebraic multiplicity. For any positive integer ''k'', the matrix ''A''<sup>''k''</sup> has as eigenvalues the powers ''x''<sub>''i''</sub><sup>''k''</sup>, and each eigenvalue <math>x_i</math> of ''A'' contributes its multiplicity to that of the eigenvalue ''x''<sub>''i''</sub><sup>''k''</sup> of ''A''<sup>''k''</sup>. Then the coefficients of the characteristic polynomial of ''A''<sup>''k''</sup> are given by the elementary symmetric polynomials ''in those powers'' ''x''<sub>''i''</sub><sup>''k''</sup>. In particular, the sum of the ''x''<sub>''i''</sub><sup>''k''</sup>, which is the ''k''-th power sum s<sub>''k''</sub> of the roots of the characteristic polynomial of ''A'', is given by its [[trace (linear algebra)|trace]]:
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| :<math>s_k = \operatorname{tr} ( A^k )\,.</math>
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| The Newton identities now relate the traces of the powers ''A''<sup>''k''</sup> to the coefficients of the characteristic polynomial of ''A''. Using them in reverse to express the elementary symmetric polynomials in terms of the power sums, they can be used to find the characteristic polynomial by computing only the powers ''A''<sup>''k''</sup> and their traces.
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| This computation requires computing the traces of matrix powers ''A''<sup>''k''</sup> and solving a triangular system of equations. Both can be done in complexity class [[NC (complexity)|NC]] (solving a triangular system can be done by divide-and-conquer). Therefore, characteristic polynomial of a matrix can be computed in NC. By the [[Cayley-Hamilton theorem]], every matrix satisfies its characteristic polynomial, and [[Cayley-Hamilton theorem#Illustration for specific dimensions and practical applications|a simple transformation]] allows to find the matrix inverse in NC.
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| Rearranging the computations into an efficient form leads to the ''[[Fadeev-Leverrier algorithm]]'' (1840), a fast parallel implementation of it is due to L. Csanky (1976). Its disadvantage is that it requires division by integers, so in general the field should have characteristic 0.
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| === Relation with Galois theory ===
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| For a given ''n'', the elementary symmetric polynomials ''e''<sub>''k''</sub>(''x''<sub>1</sub>,…,''x''<sub>''n''</sub>) for ''k'' = 1,…, ''n'' form an algebraic basis for the space of symmetric polynomials in ''x''<sub>1</sub>,…. ''x''<sub>''n''</sub>: every polynomial expression in the ''x''<sub>''i''</sub> that is invariant under all permutations of those variables is given by a [[polynomial]] expression in those elementary symmetric polynomials, and this expression is unique up to equivalence of polynomial expressions. This is a general fact known as the [[fundamental theorem of symmetric polynomials]], and Newton's identities provide explicit formulae in the case of power sum symmetric polynomials. Applied to the monic polynomial <math>\textstyle t^n+\sum_{k=1}^n (-1)^{k} a_k t^{n-k}</math> with all coefficients ''a''<sub>''k''</sub> considered as free parameters, this means that every symmetric polynomial expression ''S''(''x''<sub>1</sub>,…,''x''<sub>''n''</sub>) in its roots can be expressed instead as a polynomial expression ''P''(''a''<sub>1</sub>,…,''a''<sub>''n''</sub>) in terms of its coefficients only, in other words without requiring knowledge of the roots. This fact also follows from general considerations in [[Galois theory]] (one views the ''a''<sub>''k''</sub> as elements of a base field, the roots live in an extension field whose Galois group permutes them according to the full symmetric group, and the field fixed under all elements of the Galois group is the base field).
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| The Newton identities also permit expressing the elementary symmetric polynomials in terms of the power sum symmetric polynomials, showing that any symmetric polynomial can also be expressed in the power sums. In fact the first ''n'' power sums also form an algebraic basis for the space of symmetric polynomials.
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| == Related identities ==
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| There are a number of (families of) identities that, while they should be distinguished from Newton's identities, are very closely related to them.
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| === A variant using complete homogeneous symmetric polynomials ===
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| Denoting by ''h''<sub>''k''</sub> the [[complete homogeneous symmetric polynomial]] that is the sum of all [[monomial]]s of degree ''k'', the power sum polynomials also satisfy identities similar to Newton's identities, but not involving any minus signs. Expressed as identities of in the [[ring of symmetric functions]], they read
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| :<math>kh_k=\sum_{i=1}^kh_{k-i}p_i,</math>
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| valid for all ''k'' ≥ 1. Contrary to Newton's identities, the left-hand sides do not become zero for large ''k'', and the right-hand sides contain ever more non-zero terms. For the first few values of ''k'', one has
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| :<math>\begin{align}
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| h_1 &= p_1,\\
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| 2h_2 &= h_1p_1+p_2,\\
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| 3h_3 &= h_2p_1 + h_1p_2 + p_3.\\
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| \end{align}</math>
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| These relations can be justified by an argument analogous to the one by comparing coefficients in power series given above, based in this case on the generating function identity
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| :<math>\sum_{k=0}^\infty h_k(X_1,\ldots,X_n)t^k = \prod_{i=1}^n\frac1{1-X_it}.</math>
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| Proofs of Newton's identities, like these given below, cannot be easily adapted to prove these variants of those identities.
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| === Expressing elementary symmetric polynomials in terms of power sums ===
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| As mentioned, Newton's identities can be used to recursively express elementary symmetric polynomials in terms of power sums. Doing so requires the introduction of integer denominators, so it can be done in the ring Λ<sub>'''Q'''</sub> of symmetric functions with rational coefficients:
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| :<math>\begin{alignat}2
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| e_1 &= p_1,\\
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| e_2 &= \textstyle\frac12p_1^2 - \frac12p_2 &&= \textstyle\frac12 ( p_1^2 - p_2 ),\\
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| e_3 &= \textstyle\frac16p_1^3 - \frac12p_1 p_2 + \frac13p_3 &&= \textstyle\frac{1}{6} ( p_1^3 - 3 p_1 p_2 + 2 p_3 ),\\
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| e_4 &= \textstyle\frac1{24}p_1^4 - \frac14p_1^2 p_2 + \frac18p_2^2 + \frac13p_1 p_3 - \frac14p_4
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| &&= \textstyle\frac1{24} ( p_1^4 - 6 p_1^2 p_2 + 3 p_2^2 + 8 p_1 p_3 - 6 p_4 ),\\
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| \end{alignat}</math>
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| and so forth. Applied to a monic polynomial, these formulae express the coefficients in terms of the power sums of the roots: replace each ''e''<sub>''i''</sub> by ''a''<sub>''i''</sub> and each ''p''<sub>''k''</sub> by s<sub>''k''</sub>. | |
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| === Expressing complete homogeneous symmetric polynomials in terms of power sums ===
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| The analogous relations involving complete homogeneous symmetric polynomials can be similarly developed, giving equations
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| :<math>\begin{alignat}2
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| h_1 &= p_1,\\
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| h_2 &= \textstyle\frac12p_1^2 + \frac12p_2 &&= \textstyle\frac12 ( p_1^2 + p_2 ),\\
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| h_3 &= \textstyle\frac16p_1^3 + \frac12p_1 p_2 + \frac13p_3 &&= \textstyle\frac{1}{6} ( p_1^3 + 3 p_1 p_2 + 2 p_3 ),\\
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| h_4 &= \textstyle\frac1{24}p_1^4 + \frac14p_1^2 p_2 + \frac18p_2^2 + \frac13p_1 p_3 + \frac14p_4 | |
| &&= \textstyle\frac1{24} ( p_1^4 + 6 p_1^2 p_2 + 3 p_2^2 + 8 p_1 p_3 + 6 p_4 ),\\
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| \end{alignat}</math>
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| and so forth, in which there are only plus signs. These expressions correspond exactly to the [[cycle index]] polynomials of the [[symmetric group]]s, if one interprets the power sums ''p''<sub>''i''</sub> as indeterminates: the coefficient in the expression for ''h''<sub>''k''</sub> of any monomial ''p''<sub>1</sub><sup>''m''<sub>1</sub></sup>''p''<sub>2</sub><sup>''m''<sub>2</sub></sup>…''p''<sub>''l''</sub><sup>''m''<sub>''l''</sub></sup> is equal to the fraction of all permutations of ''k'' that have ''m''<sub>1</sub> fixed points, ''m''<sub>2</sub> cycles of length 2, …, and ''m''<sub>''l''</sub> cycles of length ''l''. Explicitly, this coefficient can be written as <math>1/N</math> where <math>N=\Pi_{i=1}^l(m_i!\,i^{m_i})</math>; this ''N'' is the number permutations commuting with any given permutation π of the given cycle type. The expressions for the elementary symmetric functions have coefficients with the same absolute value, but a sign equal to the sign of π, namely (−1)<sup>''m''<sub>2</sub>+''m''<sub>4</sub>+…</sup>.
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| === Expressing power sums in terms of elementary symmetric polynomials ===
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| One may also use Newton's identities to express power sums in terms of symmetric polynomials, which does not introduce denominators:
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| :<math>\begin{align}
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| p_1 &= e_1,\\
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| p_2 &= e_1^2 - 2 e_2,\\
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| p_3 &= e_1^3 - 3 e_2 e_1 + 3 e_3,\\
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| p_4 &= e_1^4 - 4 e_2 e_1^2 + 4 e_3 e_1 + 2 e_2^2 - 4 e_4,\\
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| p_5 &= e_1^5 - 5 e_2 e_1^3 + 5 e_3 e_1^2 + 5 e_2^2 e_1 - 5 e_4 e_1 - 5 e_3e_2 + 5 e_5,\\
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| p_6 &= e_1^6 - 6 e_2 e_1^4 + 6 e_3 e_1^3 + 9 e_2^2 e_1^2 - 6 e_4 e_1^2 - 12 e_3 e_2 e_1 + 6 e_5 e_1 - 2 e_2^3 + 3 e_3^2 + 6 e_4 e_2 - 6e_6,\\
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| \end{align}</math>
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| giving ever longer expressions that do not seem to follow any simple pattern. By consideration of the relations used to obtain these expressions, it can however be seen that the coefficient of some monomial <math>M=\Pi_{i=1}^l e_i^{m_i}</math> in the expression for <math>p_k</math> has the same ''sign'' as the coefficient of the corresponding product <math>\Pi_{i=1}^l p_i^{m_i}</math> in the expression for <math>e_k</math> described above, namely the sign (−1)<sup>''m''<sub>2</sub>+''m''<sub>4</sub>+…</sup>. Furthermore, the absolute value of the coefficient of ''M'' is the sum, over all ''distinct'' sequences of elementary symmetric functions whose product is ''M'', of the index of the last one in the sequence: for instance, the coefficient of <math>e_1^5e_3e_4^3</math> in the expression for ''p''<sub>20</sub> will be <math>(-1)^{0+3}(280\times1+56\times3+168\times4)=-1120</math>, since of all distinct orderings of the five factors ''e''<sub>1</sub>, one factor ''e''<sub>3</sub> and three factors ''e''<sub>4</sub>, there are 280 that end with ''e''<sub>1</sub>, 56 that end with ''e''<sub>3</sub>, and 168 that end with ''e''<sub>4</sub>.
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| === Expressing power sums in terms of complete homogeneous symmetric polynomials ===
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| Finally one may use the variant identities involving complete homogeneous symmetric polynomials similarly to express power sums in term of them:
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| :<math>\begin{align}
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| p_1 &= + h_1,\\
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| p_2 &= - h_1^2 + 2 h_2,\\
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| p_3 &= + h_1^3 - 3 h_2 h_1 + 3 h_3,\\
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| p_4 &= - h_1^4 + 4 h_2 h_1^2 - 4 h_3 h_1 - 2 h_2^2 + 4 h_4,\\
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| p_5 &= + h_1^5 - 5 h_2 h_1^3 + 5 h_2^2 h_1 + 5 h_3 h_1^2 - 5 h_3h_2 - 5 h_4 h_1 + 5 h_5,\\
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| p_6 &= - h_1^6 + 6 h_2 h_1^4 - 9 h_2^2 h_1^2 - 6 h_3 h_1^3 + 2 h_2^3 + 12 h_3 h_2 h_1 + 6 h_4 h_1^2 - 3 h_3^2 - 6 h_4 h_2 - 6 h_1 h_5 + 6h_6,\\
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| \end{align}</math>
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| and so on. Apart from the replacement of each ''e''<sub>''i''</sub> by the corresponding ''h''<sub>''i''</sub>, the only change with respect to the previous family of identities is in the signs of the terms, which in this case depend just on the number of factors present: the sign of the monomial <math>\Pi_{i=1}^l h_i^{m_i}</math> is −(−1)<sup>''m''<sub>1</sub>+''m''<sub>2</sub>+''m''<sub>3</sub>+…</sup>. In particular the above description of the absolute value of the coefficients applies here as well.
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| === Expressions as determinants ===
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| One can obtain explicit formulas for the above expressions in the form of determinants, by considering the first ''n'' of Newton's identities (or it counterparts for the complete homogeneous polynomials) as linear equations in which the elementary symmetric functions are known and the power sums are unknowns (or vice versa), and apply [[Cramer's rule]] to find the solution for the final unknown. For instance taking Newton's identities in the form
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| :<math>\begin{align} | |
| e_1 &= 1p_1,\\
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| 2e_2 &= e_1p_1-1p_2,\\
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| 3e_3 &= e_2p_1 - e_1p_2 + 1p_3,\\
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| \vdots &= \vdots \\
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| ne_n &= e_{n-1}p_1 - e_{n-2} p_2 + \cdots +(-1)^ne_1p_{n-1}+(-1)^{n-1}p_n\\
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| \end{align}</math>
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| we consider <math>p_1</math>, <math>{-p_2}</math>, <math>p_3</math>, ..., <math>(-1)^np_{n-1}</math> and <math>p_n</math> as unknowns, and solve for the final one, giving
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| :<math>p_n=\frac{\begin{vmatrix}1 & 0 & \cdots && e_1 \\ e_1 & 1 & 0 & \cdots & 2e_2 \\ e_2 & e_1 & 1& & 3e_3 \\ \vdots&&\ddots&\ddots&\vdots
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| \\ e_{n-1} & \cdots & e_2 & e_1 & ne_n \end{vmatrix}}{\begin{vmatrix}1 & 0 & \cdots & \\ e_1 & 1 & 0 & \cdots \\ e_2 & e_1 & 1& \\ \vdots&&\ddots&\ddots
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| \\ e_{n-1} & \cdots & e_2 & e_1 & (-1)^{n-1} \end{vmatrix}}
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| =\frac{\begin{vmatrix}1 & 0 & \cdots && e_1 \\ e_1 & 1 & 0 & \cdots & 2e_2 \\ e_2 & e_1 & 1& & 3e_3 \\ \vdots&&\ddots&\ddots&\vdots
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| \\ e_{n-1} & \cdots & e_2 & e_1 & ne_n \end{vmatrix}}{(-1)^{n-1}}
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| =\begin{vmatrix}e_1 & 1 & 0 & \cdots\\ 2e_2 & e_1 & 1 & 0 & \cdots\\ 3e_3 & e_2 & e_1 & 1 \\ \vdots &&& \ddots & \ddots
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| \\ ne_n & e_{n-1} & \cdots & & e_1 \end{vmatrix}.
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| </math>
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| Solving for <math>e_n</math> instead of for <math>p_n</math> is similar, as the analogous computations for the complete homogeneous symmetric polynomials; in each case the details are slightly messier than the final results, which are (Macdonald 1979, p. 20):
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| :<math>e_n=\frac1{n!}
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| \begin{vmatrix}p_1 & 1 & 0 & \cdots\\ p_2 & p_1 & 2 & 0 & \cdots \\ \vdots&& \ddots & \ddots \\ p_{n-1} & p_{n-2} & \cdots & p_1 & n-1 \\ p_n & p_{n-1} & \cdots & p_2 & p_1
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| \end{vmatrix},
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| \qquad p_n=(-1)^{n-1}
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| \begin{vmatrix}h_1 & 1 & 0 & \cdots\\ 2h_2 & h_1 & 1 & 0 & \cdots\\ 3h_3 & h_2 & h_1 & 1 \\ \vdots &&& \ddots & \ddots
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| \\ nh_n & h_{n-1} & \cdots & & h_1
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| \end{vmatrix},
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| \qquad h_n=\frac1{n!}
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| \begin{vmatrix}p_1 & -1 & 0 & \cdots\\ p_2 & p_1 & -2 & 0 & \cdots \\ \vdots&& \ddots & \ddots \\ p_{n-1} & p_{n-2} & \cdots & p_1 & 1-n \\ p_n & p_{n-1} & \cdots & p_2 & p_1 \end{vmatrix}.
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| </math>
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| Note that the use of determinants makes that the formula for <math>h_n</math> has additional minus signs compared to the one for <math>e_n</math>, while the situation for the expanded form given earlier is opposite. As remarked in (Littlewood 1950, p. 84) one can alternatively obtain the formula for <math>h_n</math> by taking the [[permanent]] of the matrix for <math>e_n</math> instead of the determinant, and more generally an expression for any [[Schur polynomial]] can be obtained by taking the corresponding [[immanant]] of this matrix.
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| == Derivation of the identities ==
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| Each of Newton's identities can easily be checked by elementary algebra; however, their validity in general needs a proof. Here are some possible derivations
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| === From the special case ''n'' = ''k'' ===
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| One can obtain the ''k''-th Newton identity in ''k'' variables by substitution into
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| : <math> \prod_{i=1}^k (t - x_i) = \sum_{i=0}^k (-1)^{k-i} e_{k-i}(x_1,\ldots,x_k)t^i</math>
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| as follows. Substituting ''x''<sub>''j''</sub> for ''t'' gives
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| :<math>0= \sum_{i=0}^k (-1)^{k-i} e_{k-i}(x_1,\ldots,x_k){x_j}^i \quad\text{for }1\leq j\leq k</math>
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| Summing over all ''j'' gives
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| :<math>0= (-1)^kke_k(x_1,\ldots,x_k)+\sum_{i=1}^k(-1)^{k-i} e_{k-i}(x_1,\ldots,x_k)p_i(x_1,\ldots,x_k),</math>
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| where the terms for ''i'' = 0 were taken out of the sum because ''p''<sub>0</sub> is (usually) not defined. This equation immediately gives the ''k''-th Newton identity in ''k'' variables. Since this is an identity of symmetric polynomials (homogeneous) of degree ''k'', its validity for any number of variables follows from its validity for ''k'' variables. Concretely, the identities in ''n'' < ''k'' variables can be deduced by setting ''k'' − ''n'' variables to zero. The ''k''-th Newton identity in ''n'' > ''k'' variables contains more terms on both sides of the equation than the one in ''k'' variables, but its validity will be assured if the coefficients of any monomial match. Because no individual monomial involves more than ''k'' of the variables, the monomial will survive the substitution of zero for some set of ''n'' − ''k'' (other) variables, after which the equality of coefficients is one that arises in the ''k''-th Newton identity in ''k'' (suitably chosen) variables.
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| === Comparing coefficients in series ===
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| Another derivation can be obtained by computations in the ring of [[formal power series]] ''R''[<span/>[''t'']], where ''R'' is '''Z'''[''x''<sub>1</sub>,…, ''x''<sub>''n''</sub>], the [[Polynomial ring#The polynomial ring in several variables|ring of polynomials]] in ''n'' variables ''x''<sub>1</sub>,…, ''x''<sub>''n''</sub> over the integers.
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| Starting again from the basic relation
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| : <math>\prod_{i=1}^n (t - x_i) = \sum_{k=0}^n (-1)^{k} a_k t^{n-k}</math>
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| and "reversing the polynomials" by substituting 1/''t'' for ''t'' and then multiplying both sides by ''t''<sup>''n''</sup> to remove negative powers of ''t'', gives
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| : <math>\prod_{i=1}^n (1- x_it) = \sum_{k=0}^n (-1)^{k} a_k t^k.</math>
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| (the above computation should be performed in the [[field of fractions]] of ''R''[<span/>[''t'']]; alternatively, the identity can be obtained simply by evaluating the product on the left side)
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| Swapping sides and expressing the ''a''<sub>''i''</sub> as the elementary symmetric polynomials they stand for gives the identity
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| :<math>\sum_{k=0}^n (-1)^{k} e_k(x_1,\ldots,x_n) t^k=\prod_{i=1}^n (1- x_it).</math>
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| One [[Formal derivative|formally differentiates]] both sides with respect to ''t'', and then (for convenience) multiplies by ''t'', to obtain
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| :<math>\begin{align}
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| \sum_{k=0}^n (-1)^{k}k e_k(x_1,\ldots,x_n) t^k
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| &= t \sum_{i=1}^n \left((-x_i) \prod\nolimits_{j\neq i} (1- x_jt)\right)\\
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| &= -\left(\sum_{i=1}^n \frac{x_it}{1-x_it}\right) \prod\nolimits_{j=1}^n (1- x_jt)\\
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| &= -\left(\sum_{i=1}^n \sum_{j=1}^\infty(x_it)^j\right) \left(\sum_{\ell=0}^n (-1)^\ell e_\ell(x_1,\ldots,x_n) t^\ell\right)\\
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| &= \left(\sum_{j=1}^\infty p_j(x_1,\ldots,x_n)t^j\right) \left(\sum_{\ell=0}^n (-1)^{\ell-1} e_\ell(x_1,\ldots,x_n) t^\ell\right),\\
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| \end{align}</math>
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| where the polynomial on the right hand side was first rewritten as a [[rational function]] in order to be able to factor out a product out of the summation, then the fraction in the summand was developed as a series in ''t'', using the formula
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|
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| :<math>\frac{X}{1 - X} = X + X^2 + X^3 + X^4 + X^5 + \cdots</math>,
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| and finally the coefficient of each ''t'' <sup>''j''</sup> was collected, giving a power sum. (The series in ''t'' is a formal power series, but may alternatively be thought of as a series expansion for ''t'' sufficiently close to 0, for those more comfortable with that; in fact one is not interested in the function here, but only in the coefficients of the series.) Comparing coefficients of ''t''<sup>''k''</sup> on both sides one obtains
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| :<math>(-1)^{k}k e_k(x_1,\ldots,x_n) = \sum_{j=1}^k (-1)^{k-j-1} p_j(x_1,\ldots,x_n)e_{k-j}(x_1,\ldots,x_n),</math>
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| which gives the ''k''-th Newton identity.
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| === As a telescopic sum of symmetric function identities ===
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| The following derivation, given essentially in (Mead, 1992), is formulated in the [[ring of symmetric functions]] for clarity (all identities are independent of the number of variables). Fix some ''k'' > 0, and define the symmetric function ''r''(''i'') for 2 ≤ ''i'' ≤ ''k'' as the sum of all distinct [[monomial]]s of degree ''k'' obtained by multiplying one variable raised to the power ''i'' with ''k'' − ''i'' distinct other variables (this is the [[monomial symmetric polynomial|monomial symmetric function]] ''m''<sub>γ</sub> where γ is a hook shape (''i'',1,1,…1)). In particular ''r''(''k'') = ''p''<sub>''k''</sub>; for ''r''(1) the description would amount to that of ''e''<sub>''k''</sub>, but this case was excluded since here monomials no longer have any distinguished variable. All products ''p''<sub>''i''</sub>''e''<sub>''k''−''i''</sub> can be expressed in terms of the ''r''(''j'') with the first and last case being somewhat special. One has
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| :<math>p_ie_{k-i}=r(i)+r(i+1)\quad\text{for }1<i<k</math>
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| since each product of terms on the left involving distinct variables contributes to ''r''(''i''), while those where the variable from ''p''<sub>''i''</sub> already occurs among the variables of the term from ''e''<sub>''k''−''i''</sub> contributes to ''r''(''i'' + 1), and all terms on the right are so obtained exactly once. For ''i'' = ''k'' one multiplies by ''e''<sub>0</sub> = 1, giving trivially
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| :<math>p_ke_0=p_k=r(k)\,</math>.
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| Finally the product ''p''<sub>1</sub>''e''<sub>''k''−1</sub> for ''i'' = 1 gives contributions to ''r''(''i'' + 1) = ''r''(2) like for other values ''i'' < ''k'', but the remaining contributions produce ''k'' times each monomial of ''e''<sub>''k''</sub>, since any one of the variables may come from the factor ''p''<sub>1</sub>; thus
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| :<math>p_1e_{k-1}=ke_k+r(2)\,</math>.
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| The ''k''-th Newton identity is now obtained by taking the alternating sum of these equations, in which all terms of the form ''r''(''i'') cancel out.
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| == See also ==
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| *[[Power sum symmetric polynomial]]
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| *[[Elementary symmetric polynomial]]
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| *[[Symmetric function]]
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| *[[Fluid solution]]s, an article giving an application of Newton's identities to computing the characteristic polynomial of the [[Einstein tensor]] in the case of a [[perfect fluid]], and similar articles on other types of [[exact solutions in general relativity]].
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| ==References==
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| *{{cite book | author=Tignol, Jean-Pierre | title=Galois' theory of algebraic equations | location=Singapore | publisher=World Scientific | year=2001 | isbn=978-981-02-4541-2}}
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| *{{cite book | author=Bergeron, F.; Labelle, G.; and Leroux, P. | title=Combinatorial species and tree-like structures | location=Cambridge | publisher=Cambridge University Press | year=1998 | isbn=978-0-521-57323-8}}
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| *{{cite book | author=Cameron, Peter J. | authorlink = Peter Cameron (mathematician) | title=Permutation Groups | location=Cambridge | publisher=Cambridge University Press | year=1999 | isbn=978-0-521-65378-7}}
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| *{{cite book | author=Cox, David; Little, John, and O'Shea, Donal | title=Ideals, Varieties, and Algorithms | location=New York | publisher=Springer-Verlag | year=1992 | isbn=978-0-387-97847-5}}
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| * {{cite conference | author = [[David Eppstein|Eppstein, D.]]; [[Michael T. Goodrich|Goodrich, M. T.]] | title = Space-efficient straggler identification in round-trip data streams via Newton's identities and invertible Bloom filters | booktitle = [[Workshop on Algorithms and Data Structures|Algorithms and Data Structures, 10th International Workshop, WADS 2007]] | pages = 637–648 | publisher = Springer-Verlag, Lecture Notes in Computer Science 4619 | year = 2007}} {{arxiv | 0704.3313}}
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| *{{cite book | author=Littlewood, D. E. | title=The theory of group characters and matrix representations of groups | publisher=Oxford University Press | location=Oxford | year=1950 | isbn=0-8218-4067-3 | nopp=true | page=viii+310}}
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| *{{cite book | author=Macdonald, I. G. | title=Symmetric functions and Hall polynomials | series=Oxford Mathematical Monographs | publisher=The Clarendon Press, Oxford University Press | location=Oxford | year=1979 | isbn=0-19-853530-9 | nopp=true | page=viii+180 |mr=84g:05003}}
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| *{{cite book | author=Macdonald, I. G. | title=Symmetric functions and Hall polynomials | edition=Second | series=Oxford Mathematical Monographs | publisher=Oxford Science Publications. The Clarendon Press, Oxford University Press | location=New York | year=1995 | isbn=0-19-853489-2 | page=x+475 |mr=96h:05207}}
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| *{{cite journal | author=Mead, D.G. | title=Newton's Identities | journal=The American Mathematical Monthly | volume=99 | issue=8 | year=1992-10 | pages=749–751 | doi=10.2307/2324242 | publisher=Mathematical Association of America | jstor=2324242 }}
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| *{{cite book | author=Stanley, Richard P. | authorlink = Richard P. Stanley | title=Enumerative Combinatorics, Vol. 2 | publisher=Cambridge University Press | year=1999| isbn=0-521-56069-1 (hardback), ISBN 0-521-78987-7 (paperback)}}
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| *{{cite book | author=Sturmfels, Bernd | authorlink = Bernd Sturmfels | title=Algorithms in Invariant Theory | location=New York | publisher=Springer-Verlag | year=1992 | isbn=978-0-387-82445-1}}
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| *{{cite book | author=Tucker, Alan | title=Applied Combinatorics | edition=5/e | location=New York | publisher=Wiley | year=1980 | isbn=978-0-471-73507-6}}
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| == External links ==
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| *[http://mathworld.wolfram.com/Newton-GirardFormulas.html Newton–Girard formulas] on MathWorld
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| *[http://links.jstor.org/sici?sici=0025-570X(200010)73%3A4%3C313%3AAMPONI%3E2.0.CO%3B2-0 A Matrix Proof of Newton's Identities] in Mathematics Magazine
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| *[http://stellar.mit.edu/S/course/6/sp10/6.256/courseMaterial/topics/topic2/lectureNotes/lecture-05/lecture-05.pdf Application on the number of real roots]
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| [[Category:Group theory]]
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| [[Category:Invariant theory]]
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| [[Category:Linear algebra]]
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| [[Category:Mathematical identities]]
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| [[Category:Symmetric functions]]
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| [[Category:Algebraic combinatorics]]
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| [[Category:Galois theory]]
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