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24 1. Fundamentals

1.5 Polynomials of Several Variables

If x and y are the roots of a quadratic equation at2 + bt + c = 0, then
-b/u = x + y and c/a = xy. The expressions z + y and xy are examples of
polynomials in the two variables x and y. In general, a function f(x, y) is
a polynomial in x and y if and only if it can be represented as a finite sum
of terms of the form

where c is a coefficient and k and m are nonnegative integers. The number
k + m is called the degree of the term, and the degree of the polynomial
f(x, y) is equal to the highest degree of its terms. Polynomials of several
variables can be added, subtracted and multiplied in a way analogous to
polynomials of a single variable, in which like terms are collected and the
variables are assumed to adhere to all the usual arithmetic laws.
There are two classes of polynomials of two variables which we shall
consider:

(i) symmetric polynomials f(x, y) which satisfy f(x, y) = f(y) x);

(ii) homogeneous polynomials in which all the terms are of the same

degree.

For example, si = x+y is symmetric and homogeneous of degree 1, while
s2 = xy is symmetric and homogeneous of degree 2. However, x2+x+ y+y2
is symmetric but not homogeneous, while x2y + 2x3 is homogeneous but
not symmetric.
Similar definitions can be made for functions of three variables, say x,
y, z. A polynomial is any finite sum of the type cxkymz”, with k, m, n
nonnegative integers. The degree of the polynomial is the highest degree
k + m + n of any of its terms. If all the terms have the same degree, the
polynomial is said to be homogeneous. If the polynomial f(x) y, z) satisfies

f(x, Y, 4 = f(x, z, Y> = f(y, x,4 = f(y, z,x) = f(z) 2, Y) = fk YA then
f(x, y, z) is said to be symmetric.
The elementary symmetric functions

Sl =x+y+z

s2 = xy + ye + zx

s3 = xyz

are both homogeneous and symmetric.
The purpose of this section is to introduce some elementary properties
of polynomials of several variables.