Elementary symmetric polynomials $e_k$

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Numbers

$n$

$k$

x1 + x2

x1*x2

x1 + x2 + x3

x1*x2 + x1*x3 + x2*x3

x1*x2*x3

x1 + x2 + x3 + x4

x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4

x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4

x1*x2*x3*x4

x1 + x2 + x3 + x4 + x5

x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4 + x1*x5 + x2*x5 + x3*x5 + x4*x5

x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 + x1*x2*x5 + x1*x3*x5 + x2*x3*x5 + x1*x4*x5 + x2*x4*x5 + x3*x4*x5

x1*x2*x3*x4 + x1*x2*x3*x5 + x1*x2*x4*x5 + x1*x3*x4*x5 + x2*x3*x4*x5

x1*x2*x3*x4*x5

x1 + x2 + x3 + x4 + x5 + x6

x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4 + x1*x5 + x2*x5 + x3*x5 + x4*x5 + x1*x6 + x2*x6 + x3*x6 + x4*x6 + x5*x6

x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4 + x1*x2*x5 + x1*x3*x5 + x2*x3*x5 + x1*x4*x5 + x2*x4*x5 + x3*x4*x5 + x1*x2*x6 + x1*x3*x6 + x2*x3*x6 + x1*x4*x6 + x2*x4*x6 + x3*x4*x6 + x1*x5*x6 + x2*x5*x6 + x3*x5*x6 + x4*x5*x6

x1*x2*x3*x4 + x1*x2*x3*x5 + x1*x2*x4*x5 + x1*x3*x4*x5 + x2*x3*x4*x5 + x1*x2*x3*x6 + x1*x2*x4*x6 + x1*x3*x4*x6 + x2*x3*x4*x6 + x1*x2*x5*x6 + x1*x3*x5*x6 + x2*x3*x5*x6 + x1*x4*x5*x6 + x2*x4*x5*x6 + x3*x4*x5*x6

x1*x2*x3*x4*x5 + x1*x2*x3*x4*x6 + x1*x2*x3*x5*x6 + x1*x2*x4*x5*x6 + x1*x3*x4*x5*x6 + x2*x3*x4*x5*x6

x1*x2*x3*x4*x5*x6

Definition

For $1 \leq k \leq n$ the elementary symmetric polynomial in $n$ variables is $e_k(x_1, \dots, x_n) = \sum_{i_1 < i_2 < \cdots < i_k} x_{i_1} x_{i_2} \cdots x_{i_k}$, the sum of all products of $k$ distinct variables. It has $\binom{n}{k}$ terms, each with coefficient one.

Parameters

$n$

— integer ($1 \leq n \leq 6$)

$k$

— integer ($1 \leq k \leq n$)

Formulas

(1)

$\prod_{i=1}^{n} (1 + x_i t) = \sum_{k=0}^{n} e_k(x_1,\dots,x_n)\, t^{k}$, with $e_0 = 1$.

(2)

The monic polynomial with roots $x_1, \dots, x_n$ is $\prod_i (T - x_i) = \sum_{k=0}^{n} (-1)^{k} e_k\, T^{n-k}$, which is where these are usually met.

(3)

$k\,e_k = \sum_{i=1}^{k} (-1)^{i-1} e_{k-i}\, p_i$, where $p_i$ are the power sums Power_sum_symmetric_polynomials (Newton's identities).

(4)

$\sum_{k=0}^{d} (-1)^{k} e_k\, h_{d-k} = 0$ for $d \geq 1$, where $h_d$ are the complete homogeneous symmetric polynomials Complete_homogeneous_symmetric_polynomials.

(5)

$e_k = m_{(1^k)}$, the monomial symmetric polynomial Monomial_symmetric_polynomials for the partition of $k$ into $k$ ones.

Comments

(6)

Six variables is the most this database searches: matching polynomials that differ only in the names of their variables needs a key found by trying permutations, and beyond six that is refused rather than attempted. Here it costs nothing, since $e_k$ in six variables is 217 characters written out at its longest.

(7)

$e_1, \dots, e_n$ generate the symmetric polynomials in $n$ variables freely: every symmetric polynomial is a polynomial in them in exactly one way.

Programs

(P1)

Sage

def elementary(n, k):
    R = PolynomialRing(ZZ, ['x%d' % (i+1) for i in range(n)])
    f = SymmetricFunctions(QQ).e()[k]
    return R(f.expand(n, alphabet=[str(g) for g in R.gens()]))

elementary(7, 3)                 # more variables than this table holds

Links

[1]

Wikipedia: Elementary symmetric polynomial

Data properties

Entries are of type: integral polynomial

Table is complete: false

How well the digits are known: exact