Newton identities

In mathematics , especially in algebra , the Newton identities combine two fundamental types of symmetric polynomials in a number n of variables , the elementary symmetric polynomials ${\displaystyle X_{1},\dots ,X_{n}}$

${\displaystyle \sigma _{k}(X_{1},\dots ,X_{n})=\sum _{1\leq j_{1}<\dots <j_{k}\leq n}X_{j_{1}}\cdot \ldots \cdot X_{j_{k}}}$, ${\displaystyle k=0,1,\dots ,n}$

and the power sums

${\displaystyle s_{m}(X_{1},\dots ,X_{n})=X_{1}^{m}+\ldots +X_{n}^{m}}$, ${\displaystyle m=0,1,2,\dots }$

These identities are generally traced back to the considerations of Isaac Newton around 1666, but they can also be found by Albert Girard in 1629. Applications of these identities can be found in Galois theory , invariant theory , group theory , combinatorics , but also outside of mathematics, for example in general relativity .

Derivation by means of formal power series

Let T be the variable in the ring of the formal power series . Then, analogously to Vieta's theorem , ${\displaystyle \mathbb {Q} [X_{1},\dots ,X_{n}][[T]]}$

${\displaystyle p(T)=(1+TX_{1})(1+TX_{2})\dots (1+TX_{n})=1+\sigma _{1}T+\sigma _{2}T^{2}+\dots +\sigma _{n}T^{n}}$.

Since the polynomial p (T) has a constant coefficient 1 , it can be inverted in the ring of formal power series. For the logarithmic derivation results

${\displaystyle {\frac {p'(T)}{p(T)}}={\frac {X_{1}}{1+TX_{1}}}+\dots +{\frac {X_{n}}{1+TX_{n}}}}$.

The quotients on the right also exist as formal power series; they result as geometric series . Thus applies

${\displaystyle {\frac {p'(T)}{p(T)}}=X_{1}\sum _{m=0}^{\infty }(-TX_{1})^{m}+\dots +X_{n}\sum _{m=0}^{\infty }(-TX_{n})^{m}=\sum _{m=1}^{\infty }s_{m}(-T)^{m-1}}$.

This can now be transformed into

${\displaystyle \sigma _{1}+2\sigma _{2}T+\dots +n\sigma _{n}T^{n-1}=(1+\sigma _{1}T+\dots +\sigma _{n}T^{n})\cdot (s_{1}-s_{2}T+s_{3}T^{2}-s_{4}T^{3}\pm \dots )}$.

By comparing equal powers of T on both sides, a system of equations results for determining the elementary symmetric polynomials from the power series and vice versa,

${\displaystyle {\begin{aligned}\sigma _{1}&=s_{1}\\[.3em]2\,\sigma _{2}&=s_{1}\,\sigma _{1}-s_{2}\\[.3em]3\,\sigma _{3}&=s_{1}\,\sigma _{2}-s_{2}\,\sigma _{1}+s_{3}\\[.3em]4\,\sigma _{4}&=s_{1}\,\sigma _{3}-s_{2}\,\sigma _{2}+s_{3}\,\sigma _{1}-s_{4}\\[.3em]{\text{etc.}}\qquad \\[.3em]k\,\sigma _{k}&=s_{1}\,\sigma _{k-1}-s_{2}\,\sigma _{k-2}+s_{3}\,\sigma _{k-3}\pm \ldots +(-1)^{k-2}s_{k-1}\,\sigma _{1}+(-1)^{k-1}s_{k}\\[.3em]\end{aligned}}}$

These relationships can be solved by performing the division of formal power series in p '(T) / p (T) according to the power sums, the following applies

${\displaystyle s_{1}\,=}$	${\displaystyle \sigma _{1},\,}$
${\displaystyle s_{2}\,=}$	${\displaystyle \sigma _{1}^{2}-2\,\sigma _{2},}$
${\displaystyle s_{3}\,=}$	${\displaystyle \sigma _{1}^{3}-3\,\sigma _{1}\,\sigma _{2}+3\,\sigma _{3},}$
${\displaystyle s_{4}\,=}$	${\displaystyle \sigma _{1}^{4}-4\,\sigma _{1}^{2}\,\sigma _{2}+4\,\sigma _{1}\,\sigma _{3}+2\,\sigma _{2}^{2}-4\,\sigma _{4},}$
${\displaystyle s_{5}\,=}$	${\displaystyle \sigma _{1}^{5}-5\,\sigma _{1}^{3}\,\sigma _{2}+5\,\sigma _{1}^{2}\,\sigma _{3}+5\,\sigma _{1}\,\sigma _{2}^{2}-5\,\sigma _{1}\,\sigma _{4}-5\,\sigma _{2}\,\sigma _{3}+5\,\sigma _{5},}$
${\displaystyle s_{6}\,=}$	${\displaystyle \sigma _{1}^{6}-6\,\sigma _{1}^{4}\,\sigma _{2}+6\,\sigma _{1}^{3}\,\sigma _{3}+9\,\sigma _{1}^{2}\,\sigma _{2}^{2}-6\,\sigma _{1}^{2}\,\sigma _{4}-12\,\sigma _{1}\,\sigma _{2}\,\sigma _{3}+6\,\sigma _{1}\,\sigma _{5}-2\,\sigma _{2}^{3}+6\,\sigma _{2}\,\sigma _{4}+3\,\sigma _{3}^{2}-6\,\sigma _{6},}$

Conversely, the quotient of derivative and function is the derivative of the logarithm, so after integration and application of the exponential function , the following relationships result after coefficient comparison. ${\displaystyle p(T)=\exp(s_{1}T-{\frac {1}{2}}s_{2}T^{2}+{\frac {1}{3}}s_{3}T^{3}\pm \dots )}$

${\displaystyle \sigma _{1}\,=}$	${\displaystyle s_{1},\,}$
${\displaystyle \sigma _{2}\,=}$	${\displaystyle {\frac {1}{2}}\,s_{1}^{2}-{\frac {1}{2}}\,s_{2},}$
${\displaystyle \sigma _{3}\,=}$	${\displaystyle {\frac {1}{6}}\,s_{1}^{3}-{\frac {1}{2}}\,s_{1}\,s_{2}+{\frac {1}{3}}\,s_{3},}$
${\displaystyle \sigma _{4}\,=}$	${\displaystyle {\frac {1}{24}}\,s_{1}^{4}-{\frac {1}{4}}\,s_{1}^{2}\,s_{2}+{\frac {1}{3}}\,s_{1}\,s_{3}+{\frac {1}{8}}\,s_{2}^{2}-{\frac {1}{4}}\,s_{4},}$
${\displaystyle \sigma _{5}\,=}$	${\displaystyle {\frac {1}{120}}\,s_{1}^{5}-{\frac {1}{12}}\,s_{1}^{3}\,s_{2}+{\frac {1}{6}}\,s_{1}^{2}\,s_{3}+{\frac {1}{8}}\,s_{1}\,s_{2}^{2}-{\frac {1}{4}}\,s_{1}\,s_{4}-{\frac {1}{6}}\,s_{2}\,s_{3}+{\frac {1}{5}}\,s_{5},}$
${\displaystyle \sigma _{6}\,=}$	${\displaystyle {\frac {1}{720}}\,s_{1}^{6}-{\frac {1}{48}}\,s_{1}^{4}\,s_{2}+{\frac {1}{18}}\,s_{1}^{3}\,s_{3}+{\frac {1}{16}}\,s_{1}^{2}\,s_{2}^{2}-{\frac {1}{8}}\,s_{1}^{2}\,s_{4}-{\frac {1}{6}}\,s_{1}\,s_{2}\,s_{3}+{\frac {1}{5}}\,s_{1}\,s_{5}-{\frac {1}{48}}\,s_{2}^{3}+{\frac {1}{8}}\,s_{2}\,s_{4}+{\frac {1}{18}}\,s_{3}^{2}-{\frac {1}{6}}\,s_{6}}$

literature

• Jean-Pierre Tignol: Galois's theory of algebraic equations . World Scientific, Singapore 2001, ISBN 981-02-4541-6 , doi : 10.1142 / 9789812384904 (historically oriented introduction to Galois theory).
• Peter J. Cameron: Permutation Groups . Cambridge University Press, 1999, ISBN 0-521-65378-9 (Introduction to Permutation Groups, Including Pólya's Cycle Index, Oligomorphic Permutation Groups, and Their Relationship to Mathematical Logic).
• Alan Tucker: Applied Combinatorics . Wiley, New York 1984, ISBN 0-471-86371-8 (one of the most elementary and understandable textbooks describing the Pólya enumeration formula and cycle index polynomials).