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1.5. Polynomials of Several Variables 25

Exercises

1. What are the degrees of the following polynomials? Are they homo-
   geneous? symmetric?

(4 X-Y

(b) 3xy + 2x2

(c) 4x$ + 3x + 3y + 4x2y

(d) 5x + 7

(e) xy” + yz2 + zx2

(f) xy” + yz2 + 2x2 - x2y - y2z - %2X

2. Show that a polynomial f(x, y, z) in the variables x, y, z is homoge-
   neous of degree d if and only if

f(% ty, tz) = fv(x, Y, z).

3. One can also define the notion of homogeneity for polynomials of a
   single variable. What are the homogeneous polynomials of degree k
   in a single variable x?


4. Show that each symmetric polynomial in two or three variables can
   be written as a sum of homogeneous symmetric polynomials.


5. The elementary symmetric polynomials sr = x + y and s2 = xy of
   two variables are the building blocks for all symmetric polynomials
   in the sense that every symmetric polynomial can be expressed as a
   polynomial in the elementary symmetric polynomials.
   For example,

x2 + y2 = (x + y)2 - 2xy = s: - 2s2

Prove that every symmetric polynomial can be written as a polyno-

mial of the elementary symmetric function SI, ~2.

As we have seen in Exercise 2.4, if x and y are the zeros of a quadratic

polynomial, then x + y and xy are expressible in terms of the coef-

ficients. The result just established means that we can evaluate any

symmetric polynomial function of the roots of a quadratic equation

without actually having to solve it. An analogous result for polynomi-

als of higher degree is of great practical and theoretical use, for, as we

have seen, the task of obtaining solutions to a polynomial equation

becomes heavier with the degree.