{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:MJWDPZRGUSTAPBUJ5V7DEXIE4N","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":"5783110447a10a22dea15aeb1f0288fdd5187084e313d53444ce47f4889f3057","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2017-10-09T17:48:20Z","title_canon_sha256":"a11c2888f7e2e41ec397237654545dda2c4813d39dfb45d9fb8e42e27311e3fb"},"schema_version":"1.0","source":{"id":"1710.03214","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1710.03214","created_at":"2026-05-18T00:33:26Z"},{"alias_kind":"arxiv_version","alias_value":"1710.03214v1","created_at":"2026-05-18T00:33:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.03214","created_at":"2026-05-18T00:33:26Z"},{"alias_kind":"pith_short_12","alias_value":"MJWDPZRGUSTA","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_16","alias_value":"MJWDPZRGUSTAPBUJ","created_at":"2026-05-18T12:31:31Z"},{"alias_kind":"pith_short_8","alias_value":"MJWDPZRG","created_at":"2026-05-18T12:31:31Z"}],"graph_snapshots":[{"event_id":"sha256:28a57bc1dc3e61fe1e29106b5a34b03cd8241131efa315ab0b08fea6e60247a8","target":"graph","created_at":"2026-05-18T00:33:26Z","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":"Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots $r$ is small but the multiplicities are exponentially large. Our method sets up a linear system in $r$ unknowns and iteratively builds the roots as formal power series. For an algebraic circuit $f(x_1,\\ldots,x_n)$ of size $s$ we prove that each factor has size at most a ","authors_text":"Amit Sinhababu, Nitin Saxena, Pranjal Dutta","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2017-10-09T17:48:20Z","title":"Discovering the roots: Uniform closure results for algebraic classes under factoring"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.03214","kind":"arxiv","version":1},"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:cdeeb68a02bc718f348fd3c1472324b4f92aa2859931bb2f653cc19b411141b3","target":"record","created_at":"2026-05-18T00:33:26Z","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":"5783110447a10a22dea15aeb1f0288fdd5187084e313d53444ce47f4889f3057","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2017-10-09T17:48:20Z","title_canon_sha256":"a11c2888f7e2e41ec397237654545dda2c4813d39dfb45d9fb8e42e27311e3fb"},"schema_version":"1.0","source":{"id":"1710.03214","kind":"arxiv","version":1}},"canonical_sha256":"626c37e626a4a6078689ed7e325d04e37a55e63a994b82fef49bd06ba506e2ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"626c37e626a4a6078689ed7e325d04e37a55e63a994b82fef49bd06ba506e2ff","first_computed_at":"2026-05-18T00:33:26.729084Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:33:26.729084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"I+3ND68aKsMwstDH0EKR16hyIfixFloZ3WFFg1h6ps9O7u/FzwU5p6uJl9RWJ5dHn6dyOcnQv1bcEZLWvgsjAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:33:26.729898Z","signed_message":"canonical_sha256_bytes"},"source_id":"1710.03214","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cdeeb68a02bc718f348fd3c1472324b4f92aa2859931bb2f653cc19b411141b3","sha256:28a57bc1dc3e61fe1e29106b5a34b03cd8241131efa315ab0b08fea6e60247a8"],"state_sha256":"a04d5f4986ba986bce32a67f2211f3f6df628887e5401de7e71ad2840e4ec04d"}