Supercritical discrete restriction estimates for forms in many variables

We prove discrete restriction estimates for a broad class of hypersurfaces and varieties of intermediate codimension. For our result about hypersurfaces, we use Bourgain’s arithmetic version of the Tomas–Stein method and Magyar’s decomposition of Birch’s hypersurfaces. For our result about varieties of higher codimension, we use the even moment method and works of Birch and Schmidt.