Chevalley–Warning at the boundary

The Chevalley–Warning Theorem is a result on the solution set of system polynomial equations f1,…,fr in n variables over finite field Fq low degree case d≔∑j=1rdeg(fj)<n. In this note we reformulate that terms fibers associated map and, following Heath-Brown, show something weaker continues to hold when d=n. This invites search for homogeneous polynomials which function Fqn→Fq not surjective, and exhibit several families such polynomials.