{"id":"W1669417038","doi":"10.1515/9783110250404.55","title":"Hilbert Functions of Fat Point Subschemes of the Plane: the Two-fold Way","year":2012,"lang":"en","type":"book-chapter","venue":"","topic":"Commutative Algebra and Its Applications","field":"Mathematics","cited_by":1,"is_retracted":false,"has_abstract":true,"ca_institutions":"Queen's University","funders":"","keywords":"Mathematics; Plane (geometry); Point (geometry); Pure mathematics; Mathematical analysis; Combinatorics; Geometry","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":["insufficient_payload"],"consensus_categories":[],"category_scores_codex":[0.0001924395,0.0002328279,0.0003522144,0.00004955618,0.000132442,0.00000828226,0.0004820798,0.0001254364,0.004227117],"category_scores_gemma":[0.00005752576,0.0001152702,0.0002898157,0.00004935323,0.0002625514,0.00004134345,0.0002276559,0.0003121659,0.00009403144],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00002531554,"about_ca_system_score_gemma":0.00003480848,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00006332852,"about_ca_topic_score_gemma":0.0002771874,"domain_scores_codex":[0.9989226,0.0000292209,0.0004755009,0.0001482264,0.0002798588,0.0001445899],"domain_scores_gemma":[0.9972586,0.0009563085,0.0003878973,0.001193019,0.0001659061,0.0000382362],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.000004527881,0.00004540742,0.00002469878,0.00004913177,0.0001710275,4.155895e-8,0.0004417413,3.784205e-7,0.0004417732,0.9728753,0.02525418,0.0006917824],"study_design_scores_gemma":[0.0002962641,0.00003910591,0.0002908426,0.0001499699,0.0006628777,0.000009919745,0.0003704555,0.00001304791,0.006034225,0.6951061,0.2967277,0.0002994753],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"other","genre_gemma":"other","genre_scores_codex":[0.001587913,0.0006723569,0.001521221,0.001354244,0.0001349648,0.0007605099,0.0001658841,0.0000385291,0.9937644],"genre_scores_gemma":[0.2625523,0.0001067758,0.0008661462,0.0001640146,0.0001967241,0.00005941929,0.00003013479,0.00007437001,0.7359501],"genre_candidate":"other","genre_consensus":"other","teacher_disagreement_score":0.2777691,"threshold_uncertainty_score":0.9966832,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.04866807385773432,"score_gpt":0.2802503284973015,"score_spread":0.2315822546395672,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}