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238 8. Miscellaneous Problems

property. Do there exist polynomials of higher degree which, along with all

their derivatives, have integer zeros?

E.65. Polynomials with Equally Spaced Zeros. Let f(t) be a poly-

nomial whose zeros are equally spaced, i.e. form an arithmetic progression.

Does this impose any symmetry on its graph? on the zeros of its deriva-

tives? Can anything be said in general about the relative sizes of the local

maximum and minimum values of the polynomial?

E.66. Composition of Polynomials of Several Variables. If p(t) and

q(t) are polynomials in a single variable t, then deg(p o q) = (degp)(deg q).

One consequence of this is that (p o q)(t) = t identically only if the degree

of both p and q is 1. Thus, the only polynomials of a single variable which

possess a polynomial inverse with respect to composition are linear.

What is the situation for more than one variable? Suppose, for example,

we consider the mapping

dx, Y) = (Sl(X:, Y), g2(x, Y>>

which take the real Cartesian plane of points (x, y) into itself, where the

component functions fi and gi are polynomials in the variables x and y.

These two mappings can be composed to obtain f o g:

fO!idX,Y) = (fl(sl(~:,Y),92(~,Y)),fz(gl(~,Y),g2(~,Y))).

For example, if f(x, y) = (x + y2, x3 - 22~) and g(x, y) = (xy, x - y), then

f 0 g(x, y) = (x2 + y2 - xy, X3Y3 - 2X2Y + 2XY2).

What can be said about the relationships among the degrees of the poly-
nomials? Is it necessarily the case that the degree of the components of
f o g exceeds the degree of the components of f and g. Suppose f and g
are mappings with polynomial components for which f o g(x, y) = (x, y)
identically. Must all the components off and g be of degree l?
E.67. The Mandelbrot Set. Let c be a fixed complex number and define
the quadratic polynomial f(.~) = z2 + c. Define the following sequence:

.zl = 0 r, = f(zn-l) for n 12.

on a complex plane, plot the points of this sequence for the cases c =
O,l,-l,-2, -1/2,i, 1 + i.
Depending on the value of c, the terms of the sequence can either (i)
remain within the confines of some disc of finite radius, or (ii) contain
terms of arbitrarily large absolute value. The Mandelbrot set A4 consists
of all those values of c for which (i) occurs. Using a computer or pocket
calculator, test other values of c to see whether they lie in the set M. Plot