The paucity problem for simultaneous quadratic and biquadratic equations

Abstract

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 equations

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with 0[les ]x_i, y_i[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.