{"paper":{"title":"The Arithmetic Circuit Combinatorial Nullstellensatz is NP-hard","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Andreas Bj\\\"orklund","submitted_at":"2026-06-07T14:23:23Z","abstract_excerpt":"A multivariate polynomial on $n$ variables $x_1,\\ldots,x_n$ of total degree $n$ over $\\mathbf{Z}_2$ containing the multilinear monomial $\\prod_{i=1}^n x_i$ is by the combinatorial nullstellensatz [Alon, Comb. Probab. Comput., 1999] known to always have a nonroot. We show that there cannot be a randomised polynomial time algorithm that given an arithmetic circuit of polynomial size formally computing such a polynomial, locates a nonroot with constant nonzero probability unless RP=NP. The result holds even when the individual degree of every variable in the input polynomial is at most two."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08646","kind":"arxiv","version":1},"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/2606.08646/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"}