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∑_i d_i. Next, classical Ax-Katz gives the corrected integer congruence Q^{max{0,⌈(n−h∑_i d_i)/D⌉}}| 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.

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BibTeX RIS

BibTeX

@misc{li2026from,
  title = {{From skew Combinatorial Nullstellensatz to sharp Ax-Katz congruences and zero counts}},
  author = {Shuang Li and Xuan Pang and Pingzhi Yuan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35739},
  eprint = {2610.35739},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35739},
  note = {CC BY-NC-ND 4.0},
}

RIS

TY  - UNPB
TI  - From skew Combinatorial Nullstellensatz to sharp Ax-Katz congruences and zero counts
AU  - Shuang Li
AU  - Xuan Pang
AU  - Pingzhi Yuan
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35739
UR  - https://arxiv.org/abs/2610.35739
AB  - 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∑_i d_i. Next, classical Ax-Katz gives the corrected integer congruence Q^max{0,⌈(n−h∑_i d_i)/D⌉}| 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.
ER  -

Abstract

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