In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial in one variable, they allow expressing the sums of the -th powers of all roots of (counted with their multiplicity) in terms of the coefficients of , 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.
Mathematical statement
Formulation in terms of symmetric polynomials Let be variables, denote for by the -th power sum:
and for denote by the elementary symmetric polynomial (that is, the sum of all distinct products of distinct variables), so
Then Newton's identities can be stated as
valid for all , where the left-hand size is zero for .
Concretely, one gets for the first few values of :
The form and validity of these equations do not depend on the number of variables (although the point where the left-hand side becomes 0 does, namely after the -th identity), which makes it possible to state them as identities in the ring of symmetric functions. In that ring one has
and so on; here the left-hand sides never become zero. These equations allow to recursively express the in terms of the ; to be able to do the inverse, one may rewrite them as
In general, we have
valid for all .
Also, one has
for all .
Application to the roots of a polynomial The polynomial with roots may be expanded as
where the coefficients are the symmetric polynomials defined above. Given the power sums of the roots
the coefficients of the polynomial with roots may be expressed recursively in terms of the power sums as
Formulating polynomials in this way is useful in using the method of Delves and Lyness to find the zeros of an analytic function.
Application to the characteristic polynomial of a matrix
When the polynomial above is the characteristic polynomial of a matrix (in particular when is the companion matrix of the polynomial), the roots are the eigenvalues of the matrix, counted with their algebraic multiplicity. For any positive integer , the matrix has as eigenvalues the powers , and each eigenvalue of contributes its multiplicity to that of the eigenvalue of . Then the coefficients of the characteristic polynomial of are given by the elementary symmetric polynomials in those powers . In particular, the sum of the , which is the -th power sum of the roots of the characteristic polynomial of , is given by its trace:
The Newton identities now relate the traces of the powers to the coefficients of the characteristic polynomial of . 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 and their traces.
This computation requires computing the traces of matrix powers and solving a triangular system of equations. Both can be done in complexity class 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 a simple transformation allows to find the adjugate matrix in NC.
Rearranging the computations into an efficient form leads to the Faddeev–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.
Relation with Galois theory
For a given , the elementary symmetric polynomials for form an algebraic basis for the space of symmetric polynomials in : every polynomial expression in the 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 with all coefficients considered as free parameters, this means that every symmetric polynomial expression in its roots can be expressed instead as a polynomial expression 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 as elements of a base field with roots 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).
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 power sums also form an algebraic basis for the space of symmetric polynomials.
Related identities
There are a number of (families of) identities that, while they should be distinguished from Newton's identities, are very closely related to them.
A variant using complete homogeneous symmetric polynomials
Denoting by the complete homogeneous symmetric polynomial (that is, the sum of all monomials of degree ), 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
valid for all . Contrary to Newton's identities, the left-hand sides do not become zero for large , and the right-hand sides contain ever more non-zero terms. For the first few values of , one has
Expressing elementary symmetric polynomials in terms of power sums
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 of symmetric functions with rational coefficients:
and so forth.N.b., the coefficients of the weighted product terms in the sum given by the identity above are related to the M2 numbers in Section 26.4 of the DLMF and/or the coefficients involved in the expansions of Faa di Bruno's formula The general formula can be conveniently expressed as
where the is the complete exponential Bell polynomial. This expression also leads to the following identity for generating functions:
Applied to a monic polynomial, these formulae express the coefficients in terms of the power sums of the roots: replace each by and each by .
Expressing complete homogeneous symmetric polynomials in terms of power sums
The analogous relations involving complete homogeneous symmetric polynomials can be similarly developed, giving equations
and so forth, in which there are only plus signs. In terms of the complete Bell polynomial,
These expressions correspond exactly to the cycle index polynomials of the symmetric groups, if one interprets the power sums as indeterminates: the coefficient in the expression for of any monomial is equal to the fraction of all permutations of that have fixed points, cycles of length 2, ..., and cycles of length . Explicitly, this coefficient can be written as where ; this is the number permutations commuting with any given permutation 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 , namely .
It can be proved by considering the following inductive step:
By analogy with the derivation of the generating function of the , we can also obtain the generating function of the , in terms of the power sums, as:
This generating function is thus the plethystic exponential of .
Expressing power sums in terms of elementary symmetric polynomials
One may also use Newton's identities to express power sums in terms of elementary symmetric polynomials, which does not introduce denominators:
The first four formulas were obtained by Albert Girard in 1629 (thus before Newton).
The general formula (for all positive integers ) is:
This can be conveniently stated in terms of ordinary Bell polynomials as
or equivalently as the generating function:
which is analogous to the Bell polynomial exponential generating function given in the previous subsection.
The multiple summation formula above can be proved by considering the following inductive step:
Expressing power sums in terms of complete homogeneous symmetric polynomials
Finally one may use the variant identities involving complete homogeneous symmetric polynomials similarly to express power sums in term of them:
and so on. Apart from the replacement of each by the corresponding , 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 is . In particular the above description of the absolute value of the coefficients applies here as well.
The general formula (for all non-negative integers ) is:
Expressions as determinants
One can obtain explicit formulas for the above expressions in the form of determinants, by considering the first 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
we consider and as unknowns, and solve for the final one, giving
Solving for instead of for 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):
Note that the use of determinants makes that the formula for has additional minus signs compared to the one for , 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 by taking the permanent of the matrix for instead of the determinant, and more generally an expression for any Schur polynomial can be obtained by taking the corresponding immanant of this matrix.
Derivation of the identities
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.
Proofs via generating functions We can show the relationships between , , and via the generating functions
E and H Equality of the coefficient of on each side of the equation gives us that
where we take advantage of the fact that for , and where is the Kronecker delta function, which is 1 when and is 0 otherwise. That is, the sum on the left hand side equals zero except in the trivial case that .
E and P The equation
gives us
and equality of the coefficient of on each side of that equation gives
where we have used that .
Another relationship can be found by repeatedly differentiating the right-hand side of the equation
to get its Taylor series around ; its -th term equals .
H and P Plugging
into the relationship between and gives
and equality of the coefficient of on each side of that equation gives
As a telescopic sum of symmetric function identities