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We obtain an essentially optimal estimate for the moment of order    32  /  3    of the exponential sum having argument    α   x  3   +  β   x  2    . Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine equations, one cubic and one quadratic, possess non‐trivial integral solutions whenever the number of variables exceeds   10  .

Rational solutions of pairs of diagonal equations, one cubic and one quadratic