= Newton's identities
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{title2=$m e_m=\sum_{r=1}^m(-1)^{r-1}e_{m-r}p_r$}
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= Newton-Girard identities
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{synonym}
= Newton identities
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Newton's identities let two different summaries of the same <polynomial roots> talk to each other. <Elementary symmetric polynomials> record products of distinct root positions; <power sums> record sums of individual powers, $p_r=\sum_{i=1}^n x_i^r$. Either can be easier to obtain in a particular problem. The identities turn one into the other, often avoiding any need to find the individual <polynomial roots>.
Write $e_0=1$ and $e_m=0$ for $m>n$. For every positive <integer> $m$, the identity is
$$
\boxed{m e_m=\sum_{r=1}^m(-1)^{r-1}e_{m-r}p_r}.
$$
It is a <polynomial identity> over the <integers>, so it remains true on substitution in any <commutative ring>. To solve it for $e_m$ by division, however, $m$ must have a <multiplicative inverse>. Over a <field> of <characteristic zero>, the first $n$ <power sums> recover every <elementary symmetric polynomial>. The same recovery works when the positive <characteristic of a field> is greater than $n$; small characteristic needs care.<Rational Univariate Representation (Lecture 12) by Jan Verschelde>[§4, p. 3, equations (30)–(37)]{ref}
The first three relations are $e_1=p_1$, $2e_2=e_1p_1-p_2$, and $3e_3=e_2p_1-e_1p_2+p_3$. When division by two and three is permitted, this becomes
$$
e_1=p_1,
\qquad
e_2=\frac{p_1^2-p_2}{2},
\qquad
e_3=\frac{p_1^3-3p_1p_2+2p_3}{6}.
$$
For the three roots $1,2,3$, their power sums are $p_1=6$, $p_2=14$, and $p_3=36$. The formulas give $e_2=(36-14)/2=11$ and $e_3=(216-252+72)/6=6$, recovering \b[the polynomial $z^3-6z^2+11z-6$] from aggregate data.
\Image[/newton-identities-power-sum-reconstruction.png]
{title=Three power sums reconstruct a cubic without assigning names to its roots}
{description=Left: exact contributions of the stipulated roots $1,2,3$ to the first three <power sums>. Middle: <Newton's identities> recover $e_1=6$, $e_2=11$, $e_3=6$. Right: the resulting <monic polynomial> has its three <roots of a polynomial> at the original values. These are algebraic examples, not inferred experimental data.}
The beauty is that the summaries do not depend on the order of the <polynomial roots>. In <symmetric polynomials>, one can work with quantities that survive every relabeling; in <linear algebra>, <traces> of <matrix powers> give the same kind of aggregate information about <eigenvalues>. The identities convert those summaries into <coefficients> of a <characteristic polynomial>.
A short derivation packages the identities into a <formal power series>. Let $E(t)=\prod_i(1+x_it)=\sum_{m\geq0}e_mt^m$. Since its constant coefficient is one, it has an inverse as a <formal power series>, and its <formal derivative> satisfies
$$
\frac{E'(t)}{E(t)}=\sum_i\frac{x_i}{1+x_it}=\sum_{r\geq1}(-1)^{r-1}p_rt^{r-1}.
$$
Multiply by $E(t)$ and compare the coefficient of $t^{m-1}$. This gives <Newton's identities> without a convergence assumption or division by $m$. The division issue arises only when using them to recover $e_m$. The logarithmic-derivative approach is also developed in <Newton's Identities (January 2019 notes) by Milan Mossé> (§4, pp. 6–8).
The alternate name <Newton-Girard identities> recalls <Albert Girard>'s earlier work. A modern historical account credits formulas through sixth powers to <Albert Girard> in 1629 and discusses their later appearance in <Isaac Newton>'s Arithmetica universalis.<Newton's Identities (January 2019 notes) by Milan Mossé>{ref} This citation refers to that account; the original historical editions have not been inspected here.
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