= Vinogradov mean-value method for a bilinear exponential sum
{c}
{title2=$U=\sum_{x,y\le Z}e(\sum_j\alpha_jx^jy^j)$}
Expand moments of the inner sum and group equal power-sum differences. Their multiplicities are bounded by $J_{k,r}(Z)$ through the <Cauchy-Schwarz inequality>. Two applications of the <Holder inequality> yield a bound for $|U|^{4k^2}$ containing $Z^{8k^2-4k}J_{k,r}(Z)^2$ and a product of short geometric-sum bounds over the moment differences. Thus a sharp mean-value estimate combines with rational approximation or spacing of the coefficients to prove cancellation. When all coefficients are integers, $U=\lfloor Z\rfloor^2$, so mean-value estimates alone cannot force cancellation.
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