Notes on Explorations 399and g have integer coefficients and the leading coefficient of g is 1, then it
can be arranged that h has integer coefficients.
The square problem is the special case g(t) = t2. This theorem appears
as a solution to problem E869 in the American Mathematical Monthly (56
(1949), 338; 57 (1950) 114).Th is reference also gives a history of the square
problem. See also Problems 114,190 on pages 132,143,325,341 of G. Polya
& G. Szega, Problems and Theorems in Analysis, (4th ed., Springer).
E.6. An entire set of polynomials contains at least one polynomial of each
positive degree such that any pair commute. An elementary proof that every
entire set consists, up to similarity with a linear polynomial, of either the
ordinary powers or the Tchebychef polynomials appears in H.D. Block &
H.P. Thielman, Commutative polynomials, Quart. Jour. Math. (Ser. 2) 1
(1951), 241-243.
This work is related to an attractive conjecture: Let f and g be continu-
ous functions mapping the closed unit interval {x : 0 < x C 1) into itself
for which f(g(t)) = g(f(x)) for all x; then there exists a point c for which
f(c) = g(c) = c. Is th is conjecture true for polynomials? It is refuted for
continuous functions in general by counterexamples given independently in
the papers William M. Boyce, Commuting functions with no common fixed
point, Trans. Amer. Math. Sot. 137 (1969), 77-92, MR 38 # 4267; and
John Philip Huneke, On common fixed points of commuting continuous
functions on an interval, 3’kans. Amer. Math. Sot. 139 (1969), 371-381,
MR 38 # 6005.
E.7. The connection between p,, and T, can be seen by noting that, if
x = cos 0 + i sin 0, then 2” = cos n0 + i sin n0 and t = 2 cos 0. Thus, we are
essentially interested in expressing 2 cos n0 in terms of 2 cos 0. The first few
polynomials are as follows:n p,(t)0 2
1 t
2 t2-2
3 t3 - 3t
4 t4 - 4t2 + 2 = (t2 - 2)” - 2
5 t5 - 5t3 + 5t
6 t6 - 6t4 + 9t2 - 2 = (t3 - 3t)2 - 2
= (P - 2)3 - 3(t2 - 2)
7 t7 - 7t5 + 14t3 - 7t
8 ts - 8t6 + 20t4 - 16t2 + 2
= (t4 - 4t2 + 2)2 - 2 = (t2 - 2)4 - 4(t2 - 2)2 + 2For a study of the role of these functions in determining the algebraic
character of certain values of trigonometric functions, see L. Carlitz & J.M.