{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:NIM2LAZWEFNTVS72A7Y64CFRE7","short_pith_number":"pith:NIM2LAZW","canonical_record":{"source":{"id":"1908.08759","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2019-08-23T10:59:53Z","cross_cats_sorted":[],"title_canon_sha256":"05ae7a77b5362a1fca2bfdbbbfa378fa0a2b51799532a03cef17f2fd2b6753bb","abstract_canon_sha256":"e8828dc389f1040650c4dfb0fa5f42b48c79d1775e4a28639f804cdfbd1f6488"},"schema_version":"1.0"},"canonical_sha256":"6a19a58336215b3acbfa07f1ee08b127e3a150c459712c7c977d3eb023f2b3e0","source":{"kind":"arxiv","id":"1908.08759","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.08759","created_at":"2026-07-05T02:52:37Z"},{"alias_kind":"arxiv_version","alias_value":"1908.08759v2","created_at":"2026-07-05T02:52:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08759","created_at":"2026-07-05T02:52:37Z"},{"alias_kind":"pith_short_12","alias_value":"NIM2LAZWEFNT","created_at":"2026-07-05T02:52:37Z"},{"alias_kind":"pith_short_16","alias_value":"NIM2LAZWEFNTVS72","created_at":"2026-07-05T02:52:37Z"},{"alias_kind":"pith_short_8","alias_value":"NIM2LAZW","created_at":"2026-07-05T02:52:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:NIM2LAZWEFNTVS72A7Y64CFRE7","target":"record","payload":{"canonical_record":{"source":{"id":"1908.08759","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2019-08-23T10:59:53Z","cross_cats_sorted":[],"title_canon_sha256":"05ae7a77b5362a1fca2bfdbbbfa378fa0a2b51799532a03cef17f2fd2b6753bb","abstract_canon_sha256":"e8828dc389f1040650c4dfb0fa5f42b48c79d1775e4a28639f804cdfbd1f6488"},"schema_version":"1.0"},"canonical_sha256":"6a19a58336215b3acbfa07f1ee08b127e3a150c459712c7c977d3eb023f2b3e0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:52:37.573032Z","signature_b64":"Ka2aEcsmOfSh/m3NPLSFt+/FWPA50M4vdPma1428efni4cLinuz0r8dtB70wSKViS8Nt96GUjg5GDyIcH3NbCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6a19a58336215b3acbfa07f1ee08b127e3a150c459712c7c977d3eb023f2b3e0","last_reissued_at":"2026-07-05T02:52:37.572617Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:52:37.572617Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.08759","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:52:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pXzxvj/9YqnBDVz/YOUhI9l/7/UnAH3/diArxz65MLEM0ibOJU7xFIPhQLbQjCZa/TY8sZO7+NVB9ywD6qNpAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T07:38:57.381074Z"},"content_sha256":"4bef1c07e781b77501389a5be51b00766a2813e475b0432190d150e7b081941f","schema_version":"1.0","event_id":"sha256:4bef1c07e781b77501389a5be51b00766a2813e475b0432190d150e7b081941f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:NIM2LAZWEFNTVS72A7Y64CFRE7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Number and location of pre-images under harmonic mappings in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Jan Zur, Olivier S\\`ete","submitted_at":"2019-08-23T10:59:53Z","abstract_excerpt":"We derive a formula for the number of pre-images under a non-degenerate harmonic mapping $f$, using the argument principle. This formula reveals a connection between the pre-images and the caustics. Our results allow to deduce the number of pre-images under $f$ geometrically for every non-caustic point. We approximately locate the pre-images of points near the caustics. Moreover, we apply our results to prove that for every $k = n, n+1, \\ldots, n^2$ there exists a harmonic polynomial of degree $n$ with $k$ zeros."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08759","kind":"arxiv","version":2},"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/1908.08759/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:52:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+/xVFbf/BaPEvGgbjWWfGGl3S9/EbtcPDvQEs0LyZ47Wmim4MGQ0LoQPyRxIJc8Ocl/F/aXtnubU3HQBTYr8AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T07:38:57.381582Z"},"content_sha256":"128754814f1b7ba629aa32c2885447d546fcb276ac8da1772e9d4e4cc557ca3e","schema_version":"1.0","event_id":"sha256:128754814f1b7ba629aa32c2885447d546fcb276ac8da1772e9d4e4cc557ca3e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NIM2LAZWEFNTVS72A7Y64CFRE7/bundle.json","state_url":"https://pith.science/pith/NIM2LAZWEFNTVS72A7Y64CFRE7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NIM2LAZWEFNTVS72A7Y64CFRE7/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":"2026-08-16T07:38:57Z","links":{"resolver":"https://pith.science/pith/NIM2LAZWEFNTVS72A7Y64CFRE7","bundle":"https://pith.science/pith/NIM2LAZWEFNTVS72A7Y64CFRE7/bundle.json","state":"https://pith.science/pith/NIM2LAZWEFNTVS72A7Y64CFRE7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NIM2LAZWEFNTVS72A7Y64CFRE7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:NIM2LAZWEFNTVS72A7Y64CFRE7","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":"e8828dc389f1040650c4dfb0fa5f42b48c79d1775e4a28639f804cdfbd1f6488","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2019-08-23T10:59:53Z","title_canon_sha256":"05ae7a77b5362a1fca2bfdbbbfa378fa0a2b51799532a03cef17f2fd2b6753bb"},"schema_version":"1.0","source":{"id":"1908.08759","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.08759","created_at":"2026-07-05T02:52:37Z"},{"alias_kind":"arxiv_version","alias_value":"1908.08759v2","created_at":"2026-07-05T02:52:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.08759","created_at":"2026-07-05T02:52:37Z"},{"alias_kind":"pith_short_12","alias_value":"NIM2LAZWEFNT","created_at":"2026-07-05T02:52:37Z"},{"alias_kind":"pith_short_16","alias_value":"NIM2LAZWEFNTVS72","created_at":"2026-07-05T02:52:37Z"},{"alias_kind":"pith_short_8","alias_value":"NIM2LAZW","created_at":"2026-07-05T02:52:37Z"}],"graph_snapshots":[{"event_id":"sha256:128754814f1b7ba629aa32c2885447d546fcb276ac8da1772e9d4e4cc557ca3e","target":"graph","created_at":"2026-07-05T02:52:37Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.08759/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We derive a formula for the number of pre-images under a non-degenerate harmonic mapping $f$, using the argument principle. This formula reveals a connection between the pre-images and the caustics. Our results allow to deduce the number of pre-images under $f$ geometrically for every non-caustic point. We approximately locate the pre-images of points near the caustics. Moreover, we apply our results to prove that for every $k = n, n+1, \\ldots, n^2$ there exists a harmonic polynomial of degree $n$ with $k$ zeros.","authors_text":"Jan Zur, Olivier S\\`ete","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2019-08-23T10:59:53Z","title":"Number and location of pre-images under harmonic mappings in the plane"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.08759","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":"sha256:4bef1c07e781b77501389a5be51b00766a2813e475b0432190d150e7b081941f","target":"record","created_at":"2026-07-05T02:52:37Z","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":"e8828dc389f1040650c4dfb0fa5f42b48c79d1775e4a28639f804cdfbd1f6488","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2019-08-23T10:59:53Z","title_canon_sha256":"05ae7a77b5362a1fca2bfdbbbfa378fa0a2b51799532a03cef17f2fd2b6753bb"},"schema_version":"1.0","source":{"id":"1908.08759","kind":"arxiv","version":2}},"canonical_sha256":"6a19a58336215b3acbfa07f1ee08b127e3a150c459712c7c977d3eb023f2b3e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6a19a58336215b3acbfa07f1ee08b127e3a150c459712c7c977d3eb023f2b3e0","first_computed_at":"2026-07-05T02:52:37.572617Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:52:37.572617Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ka2aEcsmOfSh/m3NPLSFt+/FWPA50M4vdPma1428efni4cLinuz0r8dtB70wSKViS8Nt96GUjg5GDyIcH3NbCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T02:52:37.573032Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.08759","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4bef1c07e781b77501389a5be51b00766a2813e475b0432190d150e7b081941f","sha256:128754814f1b7ba629aa32c2885447d546fcb276ac8da1772e9d4e4cc557ca3e"],"state_sha256":"6a34dae02d6ace825db98e24cdbaf15aa88d19005b82f219cc1370dec46b2b04"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wKQxPkl3+DNDWMFtK8Fa7R3/ARLaX2e37RAtkj57S36ActXNEI0nGAk30unzXQXyu9cxXUiZlntzeWAK6S/SDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T07:38:57.386668Z","bundle_sha256":"d559959da6e4246d79af8129a33a372d762fa5b0e3749a8ab943652a5afb447f"}}