{"paper":{"title":"Sharp Effective Finite-Field Nullstellensatz","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.AC"],"primary_cat":"math.CO","authors_text":"Guy Moshkovitz, Jeffery Yu","submitted_at":"2021-11-17T18:57:13Z","abstract_excerpt":"The (weak) Nullstellensatz over finite fields says that if $P_1,\\ldots,P_m$ are $n$-variate degree-$d$ polynomials with no common zero over a finite field $\\mathbb{F}$ then there are polynomials $R_1,\\ldots,R_m$ such that $R_1P_1+\\cdots+R_mP_m \\equiv 1$. Green and Tao [Contrib. Discrete Math. 2009, Proposition 9.1] used a regularity lemma to obtain an effective proof, showing that the degrees of the polynomials $R_i$ can be bounded independently of $n$, though with an Ackermann-type dependence on the other parameters $m$, $d$, and $|\\mathbb{F}|$. In this paper we use the polynomial method to g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.09305","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2111.09305/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}