1.5. Polynomials of Several Variables 29solutions. These formulae may be in the form of polynomials with inte-
ger coefficients. For example, verify that (X, Y,Z) = (%(x2 - y2), 2zzy,
z(z2 + y2)) satisfies the Pythagorean equation and verify that every nu-
merical solution (up to the order of X and Y) can be found by suitable
numerical substitutions for 2, y and t.
How can such polynomial solutions to diophantine equations be found?
For the Pythagorean equation, the usual argument uses some basic number
theory. But such an argument is not always readily available. Rather, one
might work empirically, using a computer to churn out a large number
of numerical solutions, and then examining these for some pattern from
which to indicate that they are values of certain polynomials. Here are
some examples for you to work on.
(a) X3 + Y3 + Z3 = W3 is satisfied by (X, Y, 2, W) =(3,4,5,6), (3,109 l&19), (4,17,22,25),
(11,15,27,29), (7,149 17,20), (12,19,53,54),
(12,31,102,103), (20,54,79,87), (23,94,105,126),
(27,46,197,198), (27,64,306,307), (28,53,75,84),
(34,39,65,72), (38,48,79,87), (48,85,491,492),
(48,109,684,685), (65,127,248,260),
(107,230,277,326), (227,230,277,356).Look for patterns which may yield solutions which are polynomials and for
which some of the numerical solutions above are obtained by evaluation of
the polynomials.
Euler generated polynomial solutions for X3 + Y3 + Z3 = W3 in the
followingway.LetX=p+q,Y=p-q,Z=r-s,W=r+s.Showthat
this leads to the requirement
p(p2 + 3q2) = s(s2 + 3r2).At this point, we introduce parameters TA, v, t, y, .z, w in such a way that u
and v appear only linearly in an equation; this will enable us to determine
their ratio in terms of z, y, z, w. Set
p = xu + 3yv s = 3zv - wu
q = yu - xv r = WV + zu.Plug these into the equation for p, q, r, s and determine what the ratio of
u to v must be. Now substitute back in to obtain expressions for p, q, r, s
and ultimately X, Y, 2, W in terms of z, y, z, w.
(b) The simultaneous system 2(B2 + 1) = A2 + C2; 2(C2 + 1) = B2 + D2