Math 2.0

Combinatorics Theorem

Newton's identities

Builds on Elementary symmetric polynomialsPower sum symmetric polynomials

Statement

Let RR be a commutative ring and let n≥1n \ge 1. Let eke_k be the elementary symmetric polynomials and pkp_k the power sums in x1,…,xnx_1, \dots, x_n, with e0=1e_0 = 1 and ek=0e_k = 0 for k>nk > n. Then for every k≥1k \ge 1, k ek=∑i=1k(−1)i−1ek−i pi.k \, e_k = \sum_{i=1}^{k} (-1)^{i-1} e_{k-i} \, p_i.

Remarks

The first cases are e1=p1e_1 = p_1, 2e2=e1p1−p22 e_2 = e_1 p_1 - p_2 and 3e3=e2p1−e1p2+p33 e_3 = e_2 p_1 - e_1 p_2 + p_3. For k>nk > n, the left side is 00, and the identity expresses pkp_k through pk−1,…,pk−np_{k-1}, \dots, p_{k-n}. Applied to the roots of a polynomial, the identities compute the power sums of the roots from the coefficients. Applied to the eigenvalues of a matrix AA, they relate the traces of the powers of AA to the coefficients of its characteristic polynomial.

History

Isaac Newton found the identities around 1666. Albert Girard had already found the first four formulas in 1629.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph
Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionPolynomials in several variables, definition, no evidence levelPolynomials in severa…definitionSymmetric polynomial, definition, no evidence levelSymmetric polynomialdefinitionElementary symmetric polynomials, definition, no evidence levelElementary symmetric …definitionPower sum symmetric polynomials, definition, no evidence levelPower sum symmetric p…definitionNewton's identities, statement, no evidence levelNewton's identitiesstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Polynomials in several variables definition
  6. Symmetric polynomial definition
  7. Elementary symmetric polynomials definition
  8. Power sum symmetric polynomials definition
  9. Newton's identities statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Newton%27s_identities, Formulation in terms of symmetric polynomials
  2. Web https://en.wikipedia.org/wiki/Newton%27s_identities, lead section
About this page
Tag0WG2A3HE0S
URIm2://0WG2A3HE0S
Aliasesgirard-newton-formulae
Kindstatement, theorem
WritingW1 stated
MSC05E05
StatusThis copy of the site has no build data. Run m2 status 0WG2A3HE0S.
UsesElementary symmetric polynomials, Power sum symmetric polynomials
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by