Smooth Hypersurfaces Beyond the Chevalley-Warning Range over Prime Fields

For every prime \(p\) , integer \(n\geq2\) , and degree \(d\geq n+1\) , we construct a geometrically smooth hypersurface \(X\subset\PP^n_{\Fp}\) of degree \(d\) such that \(\#X(\Fp)\not\equiv1\pmod p\) . The construction stays over the specified prime field in every characteristic and degree. A finite-field moment polynomial supplies nonzero coefficients with the required point count. In odd characteristic not dividing the degree, a sharper individual-degree bound permits simultaneous avoidance of the singular parameter. When the characteristic divides the degree, positive exponent compositions make a triangular family smooth for all nonzero coefficients, apart from one explicitly handled boundary case. A separate uniform formula treats characteristic two in odd degree. The coefficient selection is a finite deterministic procedure; no efficient complexity bound is asserted.