{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JOPHKGEIKHFOQTOBRYQJJKPLNA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"676e36a7ea4873ab8ede4ef99b4e9d9a24735059d26b850a02ca633a35031dd2","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-02T11:38:49Z","title_canon_sha256":"3f856106af9e63025b708080adba394b7b3280be9426ff1504dbddf0f51e4347"},"schema_version":"1.0","source":{"id":"1204.0373","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1204.0373","created_at":"2026-05-18T00:33:44Z"},{"alias_kind":"arxiv_version","alias_value":"1204.0373v2","created_at":"2026-05-18T00:33:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.0373","created_at":"2026-05-18T00:33:44Z"},{"alias_kind":"pith_short_12","alias_value":"JOPHKGEIKHFO","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JOPHKGEIKHFOQTOB","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JOPHKGEI","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:0185945e6f22ee1c77d4006daebebda9072744c38c8ac92d4a1277145368805b","target":"graph","created_at":"2026-05-18T00:33:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let p be a prime and let A=(a_1,...,a_l) be a sequence of nonzero elements in F_p. In this paper, we study the set of all 0-1 solutions to the equation a_1 x_1 + ... + a_l x_l = 0. We prove that whenever l >= p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for l < p. The critical case l=p is of particular interest. In this context, we prove that whenever l=p and A is nonconstant, the above equation has at least p-1 minimal 0-1 solutions, thus refining a theorem of Olson. The subcritical case l=p-1 is studied in detail also. Our approach is ","authors_text":"Benjamin Girard (IMJ), Eric Balandraud (IMJ)","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-02T11:38:49Z","title":"A nullstellensatz for sequences over F_p"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.0373","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:864deea6b0af45055c5dafa0bd4f2cd7892ffa50be3bb89bf13cd212e6a67f26","target":"record","created_at":"2026-05-18T00:33:44Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"676e36a7ea4873ab8ede4ef99b4e9d9a24735059d26b850a02ca633a35031dd2","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-04-02T11:38:49Z","title_canon_sha256":"3f856106af9e63025b708080adba394b7b3280be9426ff1504dbddf0f51e4347"},"schema_version":"1.0","source":{"id":"1204.0373","kind":"arxiv","version":2}},"canonical_sha256":"4b9e75188851cae84dc18e2094a9eb681c0bcdc7c5b17ad0153768f2d72f8432","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b9e75188851cae84dc18e2094a9eb681c0bcdc7c5b17ad0153768f2d72f8432","first_computed_at":"2026-05-18T00:33:44.639383Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:33:44.639383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s3HRD73kgKCvJUJosoRXIjmF7W63iOPXQROhtlyJpvSQA2O3AXmxLTUOgTNF+qmdL/Vau6Sn80AUti6Xyrt/Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:33:44.640202Z","signed_message":"canonical_sha256_bytes"},"source_id":"1204.0373","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:864deea6b0af45055c5dafa0bd4f2cd7892ffa50be3bb89bf13cd212e6a67f26","sha256:0185945e6f22ee1c77d4006daebebda9072744c38c8ac92d4a1277145368805b"],"state_sha256":"84c000eaa9d3341412f7e0523528312e8f0d0caf46fad2db16eed73e5789c91c"}