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28 1. Fundamentals

Explorations

ES. Suppose that f(x, y) is a function of the two real variables x and y.

For each llxed, value of x, f(z, y) is a polynomial in y. For each fixed value

of y, f(x, y) is a polynomial in x. Is f(x, y) necessarily a polynomial of the

two variables x and y?

There is more to this question than might seem initially apparent. The

hypothesis says, for example that, for each specific 2, f(x, y) can be written

in the form

f(X,Y) = Qo+aly+...+Q.nyn,

where not only the coefficients ai but also the degree n depends on x. On

the face of it, it might happen that for certain choices of x, n could be

arbitrarily large. However, if f(z) y) were a polynomial in x and y jointly,

the number n would not exceed some fixed number independently of x.

E.9. The Range of a Polynomial. Any polynomial is a continuous func-

tion of its variables. One important consequence is the restriction it imposes

on its possible range of values. Let f(x) be a polynomial with real coef-

ficients of n real variables, where x = (xi, x2,... , xn). For any vectors

a and b, the line segment joining a and b consists exactly of the points

(1 - t)a + tb with 0 5 t 5 1. Then p(t) = f((1 - t)a + tb) is a polynomial

in t; as t varies between 0 and 1, p(t) varies continuously between f(a) and

f(b) and accordingly assumes every value between f(a) and f(b).

For any polynomial f with real coefficients, define its range as the set

Rj = {f(x) : x = (x1 ,... , xn) with xi real}. Argue that RI must be a

subset of R of one of the following types:

(a) a singleton (i.e. a set with a single element);

(b) a finite interval with or without either endpoint;

(c) a closed halfline {r : r 5 c} or {r : r 2 c};

(d) an open halfline {r : r < c) or {r : r > c);

(e) the entire set of real numbers.

Show that (a) occurs if and only if f is a constant polynomial. Give
examples in which (c) and (e) occur. Show that (b) can never occur. Can
(d) occur for polynomials of one variable? more than one variable?

E.lO. Diophantine Equations. Who has not seen the Pythagorean equa-
tion X2 + Y2 = Z2? This is a diophantine equation with integer coefficients
and exponents and for which integer solutions are sought. This one, for ex-
ample, is satisfied by (X,Y,Z) = (3,4,5), (8, 15, 17), (5, 12, 13). Often,
diophantine equations have infinitely many solutions and the solver seeks
some formula which will give all, or at least a significant portion of, the