{"paper":{"title":"Computing Node Polynomials for Plane Curves","arxiv_id":"1006.0218","primary_cat":"math.AG","submitted_at":"2010-06-01T19:11:54+00:00","version_compiled":"v3"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13-20260917 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v3","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"According to the G\\\"ottsche conjecture (now a theorem), the degree N^{d, delta} of the Severi variety of plane curves of degree d with delta nodes is given by a polynomial in d, provided d is large enough.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.31,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":0.98,"id":"s1","own":false,"role":"background","text":"These \"node polynomials\" N_delta(d) were determined by Vainsencher and Kleiman-Piene for delta <= 6 and delta <= 8, respectively.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.05,"quantitative":true,"quantitative_p":0.95,"role_probabilities":{"gap":0.02,"method":0.0,"result":0.0,"background":0.98,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":0.85,"id":"s2","own":true,"role":"contribution","text":"Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute N_delta(d) for delta <= 14.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.97,"main_claim_p":0.52,"quantitative":true,"quantitative_p":0.94,"role_probabilities":{"gap":0.0,"method":0.15,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.85,"interpretation":0.0}},{"i":3,"p":0.71,"id":"s3","own":true,"role":"result","text":"Furthermore, we improve the threshold of polynomiality and verify G\\\"ottsche's conjecture on the optimal threshold up to delta <= 14.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.24,"quantitative":true,"quantitative_p":0.94,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.71,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.29,"interpretation":0.0}},{"i":4,"p":0.75,"id":"s4","own":true,"role":"result","text":"We also determine the first 9 coefficients of N_delta(d), for general delta, settling and extending a 1994 conjecture of Di Francesco and Itzykson.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.24,"quantitative":true,"quantitative_p":0.96,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.75,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.25,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.84,"text":"Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute N_delta(d) for delta <= 14.","location":"abstract s2","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"OpenRouter","model":"classifier-1.13","endpoint":"https://openrouter.ai/api/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":4400,"cost_usd_reported":0.000185},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13-20260917","abstract_states":{"claims_first":0.13,"states_limitation":0.54,"human_participants":0.01,"controlled_comparison":0.02,"randomized_assignment":0.01,"states_quantitative_result":0.96,"compares_with_prior_results":0.93,"mentions_code_or_data_availability":0.03},"evidential_basis":{"p":0.66,"value":"simulation or computation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":0.34,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.66}},"contribution_type":{"p":0.89,"value":"method or tool","probabilities":{"review":0.0,"method_or_tool":0.89,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":0.11,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":1.0,"text":"Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute N_delta(d) for delta <= 14.","location":"abstract s2"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v3"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":5,"sentences_in_abstract":5},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; 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