{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2008:UCODSUPLBXBNUL5IDKAEZHHSGZ","short_pith_number":"pith:UCODSUPL","canonical_record":{"source":{"id":"0812.4193","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-12-22T14:10:23Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"e2ec986a2137bed36756ee2bd5542085e50bdb9369e184aa074182827307f0d6","abstract_canon_sha256":"2a6581aa8871e039cb8cb7e957eb0fdc07cb4585fa7339dbea08392262565ebc"},"schema_version":"1.0"},"canonical_sha256":"a09c3951eb0dc2da2fa81a804c9cf2365b4580d2a986f59c8aae514a79f5f430","source":{"kind":"arxiv","id":"0812.4193","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0812.4193","created_at":"2026-05-18T02:58:10Z"},{"alias_kind":"arxiv_version","alias_value":"0812.4193v1","created_at":"2026-05-18T02:58:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0812.4193","created_at":"2026-05-18T02:58:10Z"},{"alias_kind":"pith_short_12","alias_value":"UCODSUPLBXBN","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"UCODSUPLBXBNUL5I","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"UCODSUPL","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2008:UCODSUPLBXBNUL5IDKAEZHHSGZ","target":"record","payload":{"canonical_record":{"source":{"id":"0812.4193","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-12-22T14:10:23Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"e2ec986a2137bed36756ee2bd5542085e50bdb9369e184aa074182827307f0d6","abstract_canon_sha256":"2a6581aa8871e039cb8cb7e957eb0fdc07cb4585fa7339dbea08392262565ebc"},"schema_version":"1.0"},"canonical_sha256":"a09c3951eb0dc2da2fa81a804c9cf2365b4580d2a986f59c8aae514a79f5f430","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:10.862797Z","signature_b64":"4uH46xCsjHaa1EB6GTC1+Lc38a2O7NHFxdgUR1vbI8ikaszoFSISnM3hcpTgHl0ME0671+E95QGNDdUTt+FgBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a09c3951eb0dc2da2fa81a804c9cf2365b4580d2a986f59c8aae514a79f5f430","last_reissued_at":"2026-05-18T02:58:10.862345Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:10.862345Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0812.4193","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U6fg5rlx7EPyjvkAFFnuGGUE3vUmDytypCwYTZbCIEIL23WpzvNq0EZk+TyouJuZCUvLZCU5/P3vQHDyr5uEDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:38:53.578853Z"},"content_sha256":"7c4c535874e40299fe9cda6261f25d300a6ee9dc9778b8ee1170bfc22e4fa636","schema_version":"1.0","event_id":"sha256:7c4c535874e40299fe9cda6261f25d300a6ee9dc9778b8ee1170bfc22e4fa636"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2008:UCODSUPLBXBNUL5IDKAEZHHSGZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algebro-gemetric aspects of Heine-Stieltjes theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Boris Shapiro","submitted_at":"2008-12-22T14:10:23Z","abstract_excerpt":"The goal of this paper is to develop a Heine-Stieltjes theory for univariate linear differential operators of higher order. Namely, for a given given operator T=\\sum_i Q_i(z)d^i/dz^i with polynomial coefficients Q_i(z) set r=max_i (deg Q_i(z)-i). Following the classical approach of Heine and Stieltjes we study the multiparameter spectral problem of finding all polynomial V(z) of degree at most r such that the equation:\n  T(z)S(z)+V(z)S(z=0 has for a given positive integer n a polynomial solution S(z) of degree n. We show that under some mild non-degeneracy assumptions there exist exactly ((n+r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0812.4193","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"neoMnjJsGv8OJbMzbI+CYPJW5XSgo7zPi6tjY2PmTst0qPkSB6WkeJiZ+umdrxmAz3z+sVQcEZ6bx2QcSDdTBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:38:53.579542Z"},"content_sha256":"00c63abcdd8c2a1005b1077a75366483b1951d23038ae7e8c8e7da22e9a78ba2","schema_version":"1.0","event_id":"sha256:00c63abcdd8c2a1005b1077a75366483b1951d23038ae7e8c8e7da22e9a78ba2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UCODSUPLBXBNUL5IDKAEZHHSGZ/bundle.json","state_url":"https://pith.science/pith/UCODSUPLBXBNUL5IDKAEZHHSGZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UCODSUPLBXBNUL5IDKAEZHHSGZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T20:38:53Z","links":{"resolver":"https://pith.science/pith/UCODSUPLBXBNUL5IDKAEZHHSGZ","bundle":"https://pith.science/pith/UCODSUPLBXBNUL5IDKAEZHHSGZ/bundle.json","state":"https://pith.science/pith/UCODSUPLBXBNUL5IDKAEZHHSGZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UCODSUPLBXBNUL5IDKAEZHHSGZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2008:UCODSUPLBXBNUL5IDKAEZHHSGZ","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":"2a6581aa8871e039cb8cb7e957eb0fdc07cb4585fa7339dbea08392262565ebc","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-12-22T14:10:23Z","title_canon_sha256":"e2ec986a2137bed36756ee2bd5542085e50bdb9369e184aa074182827307f0d6"},"schema_version":"1.0","source":{"id":"0812.4193","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0812.4193","created_at":"2026-05-18T02:58:10Z"},{"alias_kind":"arxiv_version","alias_value":"0812.4193v1","created_at":"2026-05-18T02:58:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0812.4193","created_at":"2026-05-18T02:58:10Z"},{"alias_kind":"pith_short_12","alias_value":"UCODSUPLBXBN","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"UCODSUPLBXBNUL5I","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"UCODSUPL","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:00c63abcdd8c2a1005b1077a75366483b1951d23038ae7e8c8e7da22e9a78ba2","target":"graph","created_at":"2026-05-18T02:58:10Z","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":"The goal of this paper is to develop a Heine-Stieltjes theory for univariate linear differential operators of higher order. Namely, for a given given operator T=\\sum_i Q_i(z)d^i/dz^i with polynomial coefficients Q_i(z) set r=max_i (deg Q_i(z)-i). Following the classical approach of Heine and Stieltjes we study the multiparameter spectral problem of finding all polynomial V(z) of degree at most r such that the equation:\n  T(z)S(z)+V(z)S(z=0 has for a given positive integer n a polynomial solution S(z) of degree n. We show that under some mild non-degeneracy assumptions there exist exactly ((n+r","authors_text":"Boris Shapiro","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-12-22T14:10:23Z","title":"Algebro-gemetric aspects of Heine-Stieltjes theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0812.4193","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":"sha256:7c4c535874e40299fe9cda6261f25d300a6ee9dc9778b8ee1170bfc22e4fa636","target":"record","created_at":"2026-05-18T02:58:10Z","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":"2a6581aa8871e039cb8cb7e957eb0fdc07cb4585fa7339dbea08392262565ebc","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2008-12-22T14:10:23Z","title_canon_sha256":"e2ec986a2137bed36756ee2bd5542085e50bdb9369e184aa074182827307f0d6"},"schema_version":"1.0","source":{"id":"0812.4193","kind":"arxiv","version":1}},"canonical_sha256":"a09c3951eb0dc2da2fa81a804c9cf2365b4580d2a986f59c8aae514a79f5f430","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a09c3951eb0dc2da2fa81a804c9cf2365b4580d2a986f59c8aae514a79f5f430","first_computed_at":"2026-05-18T02:58:10.862345Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:10.862345Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4uH46xCsjHaa1EB6GTC1+Lc38a2O7NHFxdgUR1vbI8ikaszoFSISnM3hcpTgHl0ME0671+E95QGNDdUTt+FgBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:10.862797Z","signed_message":"canonical_sha256_bytes"},"source_id":"0812.4193","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7c4c535874e40299fe9cda6261f25d300a6ee9dc9778b8ee1170bfc22e4fa636","sha256:00c63abcdd8c2a1005b1077a75366483b1951d23038ae7e8c8e7da22e9a78ba2"],"state_sha256":"c69725236f85a0d3c57e545db90d434a57fc89b924c03f8440676aca2f15f62b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I1NOTlvJcg8z+Mwmt+KZVDqUIj2JZ0YM6AabXCQkCjG/siodoqGmDjwyutv9thdQVjtIY/Je1QeVHBOySgOoAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T20:38:53.582449Z","bundle_sha256":"baf4f2f86e43b6e0fdccb74cb8e3d72c25e6e21f0e9e2ab721d225380598dab0"}}