1.5. Polynomials of Several Variables 27
- A polynomial of several variables tl, t2,... , t, is a finite sum of mono-
mials of the type __
+‘2a... trm m
where a is a coefficient and the ri are nonnegative integers. The degree
ofthis term is rl+r2+rg+...+r,,,, and the degree of the polynomial
is equal to the highest degree of any of its terms.
A polynomial of several variables is homogeneous (of degree d) if and
only if each term is of the same degree (d).
A polynomial of several varibles is symmetric if it remains unchanged
no matter how we interchange its variables.
Given a set of variables, tl, t;! ,... , t,, there is a special class of sym-
metric polynomials associated with them. There are the elementary
symmetric functions:
Sl =sl(tl,t2,...,t,)=tl+t2+***+t,s2 = SZ(tllt2,... ,tm) = t1t2 -l-t1t3 + --*+t1t,
+ t&l +... + t,-1t,...
s, is the sum of all possible products of P of the variables (this sum
terms)...
s,-1 = Sm-l(tl,t2 ,...) tm) = -&t2...i,...tm
i=l
(A “hat” denotes a deleted term.)Give all symmetric homogeneous polynomials of degree 0, 1 and 2 in
the variables tl, t2, t3,... , t,, and show how they can be expressed
as a polynomial in the functions sr and ~2.- Formulate and prove the analogue of Exercise 2 for any number of
variables. - The polynomial g(x, y) has the property that, for any numerical sub-
stitutions of x and y, g(z, y) = g(y, x). Must g(z, y) be symmetric in
the variables x and y?