Notes on Explorations 415, (i) m = 2, n is even;
(ii) n = 2, m is arbitrary;(iii) m = 3, n = 4.See Section 2.23 of the above reference for a brief discussion.
Research on expressing polynomials as sums of squares has recently be-
come active. See, for example, the paper M.-D. Choi & T.-Y. Lam, An
old question of Hilbert, Conference on quadratic forms, 1976 (G. Orzech,
ed.), Queen’s Papers in Pure and Applied Mathematics, 46 (1977), 385-405
(Queen’s University, Kingston, Ontario).
E.62. It is readily verified thatQ3(%) = 3(1+ %)q4(z) = 4(2 + 3z + 2z2)Q5(%) = 5(1+ z)(l + % + 2”)
Q7(%) = 7(1+ %)(l + % + %2)2
all have zeros on the unit circle. When n 2 8, q;(z) has a zero whose
absolute values exceeds 1, so that, by the Gauss-Lucas Theorem, the same
is true of qn(z) itself. See Problem E3078 in Amer. Math. Monthly 92
(1985), 215; 95 (1988)) 140. As for q,,(6), it can be written in the form
(a+bt+az2)(c+dz+cz2) w h ere ad and bc are the zeros of the quadratic
t2 - 15t + 48. From this it follows that @j(%) has two real quadratic factors
and it can be further shown that each has nonreal zeros, so that all the
zeros of &j(Z) lies on the unit circle.
E.63. For an example of a polynomial mapping in two variables with poly-
nomial inverse, consult page 694 of Gary H. Meisters, Jacobian problems
in differential equations and algebraic geometry, Rocky Mountain J. Math.
12 (1982) 679-705. F or an indication of the significance of this question
in the study of differential equations and for further references, see Hyman
Bass & Gary Meisters, Polynomial flows in the plane, Advances in Math.
55 (1985), 173-208.
E.67. For an elementary introduction to the Mandelbrot set, see the follow-
ing Computer Recreations columns by A.K. Dewdney in Scientific Ameri-
can: 253 (no. 2)) 198, 16-24; 257 ( no. 5), 1987, 140-145. See the note on
Exploration E.49 for related references.