In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rational coefficients. However, not every symmetric polynomial with integral coefficients is generated by integral combinations of products of power-sum polynomials: they are a generating set over the rationals, but not over the integers.
The power sum symmetric polynomial of degree k in
n
pk(x1,x2,...,xn)=
n | |
\sum | |
i=1 |
k | |
x | |
i |
.
p0(x1,x2,...,xn)=1+1+ … +1=n,
p1(x1,x2,...,xn)=x1+x2+ … +xn,
p2(x1,x2,...,xn)=
2 | |
x | |
1 |
+
2 | |
x | |
2 |
+ … +
2 | |
x | |
n |
,
p3(x1,x2,...,xn)=
3 | |
x | |
1 |
+
3 | |
x | |
2 |
+ … +
3 | |
x | |
n |
.
k
k
n
The polynomial ring formed by taking all integral linear combinations of products of the power sum symmetric polynomials is a commutative ring.
The following lists the
n
n.
p0=n
For n = 1:
p1=x1.
For n = 2:
p1=x1+x2,
p2=
2 | |
x | |
1 |
+
2. | |
x | |
2 |
For n = 3:
p1=x1+x2+x3,
p2=
2 | |
x | |
1 |
+
2 | |
x | |
2 |
+
2, | |
x | |
3 |
p3=
3, | |
x | |
3 |
The set of power sum symmetric polynomials of degrees 1, 2, ..., n in n variables generates the ring of symmetric polynomials in n variables. More specifically:
Theorem. The ring of symmetric polynomials with rational coefficients equals the rational polynomial ring
Q[p1,\ldots,pn].
However, this is not true if the coefficients must be integers. For example, for n = 2, the symmetric polynomial
P(x1,x2)=
2 | |
x | |
1 |
x2+x1
2 | |
x | |
2 |
+x1x2
P(x1,x2)=
| ||||||||||
2 |
+
| |||||||
2 |
,
P(x1,x2)
Z[p1,\ldots,pn].
e2:=\sum1xixj=
| |||||||
2 |
.
The theorem is also untrue if the field has characteristic different from 0. For example, if the field F has characteristic 2, then
p2=
2 | |
p | |
1 |
Sketch of a partial proof of the theorem: By Newton's identities the power sums are functions of the elementary symmetric polynomials; this is implied by the following recurrence relation, though the explicit function that gives the power sums in terms of the ej is complicated:
pn=
n | |
\sum | |
j=1 |
(-1)j-1ejpn-j.
Rewriting the same recurrence, one has the elementary symmetric polynomials in terms of the power sums (also implicitly, the explicit formula being complicated):
en=
1 | |
n |
n | |
\sum | |
j=1 |
(-1)j-1en-jpj.
This implies that the elementary polynomials are rational, though not integral, linear combinations of the power sum polynomials of degrees 1, ..., n. Since the elementary symmetric polynomials are an algebraic basis for all symmetric polynomials with coefficients in a field, it follows that every symmetric polynomial in n variables is a polynomial function
f(p1,\ldots,pn)
Q[p1,\ldots,pn].
(This does not show how to prove the polynomial f is unique.)
For another system of symmetric polynomials with similar properties see complete homogeneous symmetric polynomials.