From skew Combinatorial Nullstellensatz to sharp Ax-Katz congruences and zero counts

Abstract

We give a unified account of zero counting in finite-field polynomial rings with commuting variables twisted by a common automorphism. A decomposition into fixed-field slices connects skew evaluation with ordinary polynomial evaluation. First, we derive a slice coefficient formula from the skew Combinatorial Nullstellensatz and use it to deduce skew Chevalley-Warning, including the sufficient condition $n>h\sum_i d_i$. Next, classical Ax-Katz gives the corrected integer congruence $Q^{\max\{0,\lceil(n-h\sum_i d_i)/D\rceil\}}\mid N+(M-1)δ$. We construct homogeneous examples proving optimality for every positive degree table and every number of variables. We also establish a sharp lower bound with a prescribed zero, an input-sensitive coefficient-filtration refinement, and transfers of known bounds for nonlinear zero sets. Detailed calculations explain the origin correction, incomplete norm blocks, and the distinction between characteristic-$p$ summation and higher integer divisibility. Existing geometric, interpolation, and counting theorems are explicitly attributed.