The problem of constructing non-diagonal solutions to systems of
symmetric
diagonal equations has attracted intense investigation for centuries
(see [5, 6] for a
history of such problems) and remains a topic of current interest (see,
for example,
[2–4]). In contrast, the problem of bounding
the number of such non-diagonal
solutions has commanded attention only comparatively recently, the first
non-trivial
estimates having been obtained around thirty years ago through the sieve
methods
applied by Hooley [10, 11] and Greaves
[7] in their investigations concerning sums
of two kth powers. As a further contribution to the problem of
establishing the
paucity of non-diagonal solutions in certain systems of diagonal diophantine
equations, in this paper we bound the number of non-diagonal solutions
of a system
of simultaneous quadratic and biquadratic equations. Let
S(P) denote the number of
solutions of the simultaneous diophantine equationsformula herewith 0[les ]xi,
yi[les ]P(1[les ]i[les ]3),
and let
T(P) denote the corresponding number of
solutions with (x1, x2, x3)
a permutation of (y1, y2, y3).
In Section 4 below we establish the upper and lower bounds for
S(P)−T(P) contained in the following
theorem.
Solving dynamic discrete choice models using smoothing and sieve methods
