Website owner: James Miller
Elementary symmetric polynomials. Viete’s Formulas.
Def. Symmetric polynomial. A function of two or more variables which remains unchanged under every interchange of two of the variables.
Example. xy + xz + yz
Def. Elementary symmetric polynomials. For the case of three variables x1, x2, x3, the elementary symmetric polynomials are
1) σ1 = x1 + x2 + x3
σ2 = x1 x2 + x1 x3 + x2 x3
σ3 = x1 x2 x3
which arise in the expansion
2) (t - x1)(t - x2)(t - x3) = t3 - σ1t2 + σ2t - σ3
For the case of four variables x1, x2, x3, x4 the elementary symmetric polynomials are
3) σ1 = x1 + x2 + x3 + x4
σ2 = x1 x2 + x1 x3 + x1 x4 + x2 x3 + x2 x4 + x3 x4
σ3 = x1 x2 x3 +x1 x2 x4 + x1 x3 x4 + x2 x3 x4 (i.e. sum of all combinations of the variables x1, x2, ... , xn taken 3 at a time)
σ4 = x1 x2 x3 x4
For the case of n variables x1, x2, ... , xn, the elementary symmetric polynomials are
4) σ1 = x1 + x2 + ... + xn
.......................
σk = x1 x2 x3 ... + x1 x3 x4 ... + ... (sum of all combinations of the variables x1, x2, ... , xn taken k at a time)
.......................
σn = x1 x2 x3 ... xn
which arise in the expansion
4) (t - x1)(t - x2) ... (t - xn) = tn - σ1tn-1 + σ2tn-2 - σ2tn-3 + ... + (-1)nσn
Theorem. Any symmetric polynomial p(x1, x2, ... , xn) can be expressed as a polynomial in the elementary symmetric polynomials.
Examples.
1. x2 + y2 = (x + y)2 - 2xy = σ12 - 2σ2 where σ1 = x + y and σ2 = xy
2. x3 + y3 = (x + y)3 - 3xy(x + y) = σ1(σ12 - 3σ2) = σ13 - 3σ1σ2 where σ1 = x + y and σ2 = xy
[Note. It can be shown through expansion that x3 + y3 = (x + y)3 - 3xy(x + y) ]
Def. The group of the polynomial. In the polynomial p(x1, x2, ... , xn), the set of all those permutations of the indices which leave the polynomial unchanged.
Viete’s Formulas. Let x1, x2, ... , xn be the n roots of the polynomial
f(x) = xn + a1xn-1 + a2xn-2 + ... + an
where the coefficients a1, a2, ..., an are real or complex numbers. Then
f(x) = (x - x1)(x - x2) ... (x - xn)
where x1, x2, ..., xn are real or complex numbers.
If we multiply out the expression (x - x1)(x - x2) ... (x - xn) we find
-a1 = x1 + x2 + ... + xn
a2 = x1 x2 + x1 x3 + x1 x4 + ... (sum of all combinations of the variables x1, x2, ... , xn taken 2 at a time)
-a3 = x1 x2 x3 + x1 x2 x4 + x1 x3 x4 ... (sum of all combinations of the variables x1, x2, ... , xn taken 3 at a time)
............................................................
-ak = x1 x2 x3... + x1 x3 x4 ... + x1 x3 x5 ... (sum of all combinations of the variables x1, x2, ... , xn taken k at a time)
............................................................
(-1)nan = x1 x2 x3 ... xn
Thus
a1 = -σ1
a2 = σ2
a3 = -σ3
..........
an = (-1)nσn
Note. Note that the function
f(x1, x2, ... , xn) = (x - x1)(x - x2) ... (x - xn)
is a symmetric function when regarded as a function of the roots x1, x2, ... , xn with x held constant.
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