Weyl inequality ($|S|{\ll }_{k,\eta }{N}^{1+\eta }(q^{-1}+N^{-1}+q{N}^{-k}{)}^{2^{1-k}}$)

For a real degree-$k$ polynomial with leading coefficient $\theta$, $k\ge 2$, and coprime integers $a,q$ satisfying $|\theta -a/q|\le q^{-2}$, the displayed bound applies to its exponential sum over $1\le n\le N$. Repeated Weyl differencing reduces the phase to slope $k!\theta h_1\cdots h_{k-1}$. A geometric-sum estimate, the subpower bound for the divisor function and the reciprocal fractional-part sum near a rational complete the estimate. The arbitrary positive exponent $\eta$ absorbs divisor bounds and logarithms.