Unknown

60 2. Evaluation, Division, and Expansion

(a) Set z = 0 and obtain that f (x, y, 0) = (x + y)x29.

(b) Consider the polynomial

Ll(x, Y, z) = f(x, Y, z) - 31(x, YI Z>ES2(X, Y, 412.

Show that g(x, y, 0) = g(x, 0, y) = g(0, y, z) = 0 and deduce that

xyz is a factor of g(x, y, z).

(c) Determine a polynomial h(x, y, z) for which

dx, Y, 4 = XYZ h(x, y, 4.

Is h(x, y, z) symmetric and homogeneous? What is the degree of

h(x, Y, +’

(d) Write h(x, y, z ) as a polynomial in si, s2 and ss.

(e) Write f (x, y, z) as a polynomial in si, $2 and ss.

14. Carry out the procedure of Exercise 13 on the other two polynomials
    given in Exercise 1.5.8.

15: -Gauss’ Theorem on Symmetric Functions. In Exercises 1.5.5 and

s. , 1.5.8, the representation of a symmetric polynomial in terms of the
elementary symmetric polynomials was carried out for specific exam-
ples of low degree. In this exercise, we will outline the proof of this
result in general.
Lettl,tz,... , t, be n variables and let sl,s2,... , s,, be the elementary
symmetric functions of these variables; namely, si is the sum of the
n
( i >

possible products of i of the variables tk (1 5 j 5 n). Then

any symmetric polynomial in the variables 6 can be ex-

pressed as a polynomial in the variable si (1 5 i 5 n).

(a) It is enough to prove the result for homogeneous polynomials.

We use induction. Verify that, trivially, the result holds for all

polynomials of degree 0 and for all polynomials of a single vari-

able.

(b) Suppose as an induction hypothesis, that the result holds for

(i) all polynomials of degree < k and any number of variables;

(ii) all polynomials of degree k and n - 1 or fewer variables;

where k 2 1 and n 2 2.

Let p(h,t2,t3,... , tn) be a homogeneous symmetric polynomial

of degree k. Show that p(tl,tz,... ,tn-i, 0) is a homogeneous

symmetric polynomial of n - 1 variables which, by the induction

hypothesis, can be written in the form q(ul, 212,... , un-l), where

q is a polynomial in the elementary symmetric functions uj of

the n - 1 variables ti (1 < i 5 n - 1).