{"paper":{"title":"The Cullis' determinant as Pfaffian","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The Cullis determinant of a rectangular matrix equals the Pfaffian of a skew-symmetric matrix obtained from it by multiplication and transposition.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Guterman, Andrey Yurkov","submitted_at":"2026-05-13T18:21:43Z","abstract_excerpt":"The Cullis' determinant is a generalization of the ordinary determinant for rectangular matrices. It is defined as the alternating sum of maximal minors of given matrix. In this paper we express the Cullis' determinant of a matrix $X$ as the Pfaffian of the matrix obtained from $X$ by matrix multiplication and transposition.\n  Relying on this result, we present an efficient polynomial-time algorithm for calculating the Cullis' determinant of given matrix."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we express the Cullis' determinant of a matrix X as the Pfaffian of the matrix obtained from X by matrix multiplication and transposition","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The specific matrix construction via multiplication and transposition produces a skew-symmetric matrix whose Pfaffian exactly matches the alternating sum of maximal minors for arbitrary rectangular X.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Cullis' determinant of rectangular matrix X equals the Pfaffian of a matrix obtained from X by multiplication and transposition, enabling an efficient polynomial-time algorithm.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Cullis determinant of a rectangular matrix equals the Pfaffian of a skew-symmetric matrix obtained from it by multiplication and transposition.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"bd0009cedd5354e35d3e2cdbda6862291fb7830fc6348b5536b9b7a449588067"},"source":{"id":"2605.14010","kind":"arxiv","version":1},"verdict":{"id":"8768bc25-71be-4f8a-857e-c8b0977d8fed","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:49:03.614993Z","strongest_claim":"we express the Cullis' determinant of a matrix X as the Pfaffian of the matrix obtained from X by matrix multiplication and transposition","one_line_summary":"The Cullis' determinant of rectangular matrix X equals the Pfaffian of a matrix obtained from X by multiplication and transposition, enabling an efficient polynomial-time algorithm.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The specific matrix construction via multiplication and transposition produces a skew-symmetric matrix whose Pfaffian exactly matches the alternating sum of maximal minors for arbitrary rectangular X.","pith_extraction_headline":"The Cullis determinant of a rectangular matrix equals the Pfaffian of a skew-symmetric matrix obtained from it by multiplication and transposition."},"references":{"count":20,"sample":[{"doi":"","year":2010,"title":"A. Amiri, M. Fathy, and M. Bayat. Generalization of some determinantal identities for non-square matrices based on Radic’s definition.TWMS J. Pure Appl. Math., 1(2):163– 175, 2010","work_id":"f96ad504-3b62-4386-9ad8-cd863cea4d7e","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Dover Books on Mathematics","work_id":"4400e67a-cabd-4996-a0ed-247530a111f8","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1913,"title":"C. E. Cullis.Matrices and Determinoids: Volume 1. Calcutta University Readership Lectures. Cambridge University Press, 1913","work_id":"64609c2d-e325-426e-97b4-501c2e609717","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"W. Fulton and P. Pragacz.Schubert Varieties and Degeneracy Loci. Springer Berlin Heidelberg, 1998","work_id":"150749e3-edb9-41a3-9f9e-6ed05d7dcf3d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1994,"title":"G. Galbiati and F. Maffioli. On the computation of pfaffians.Discrete Applied Mathe- matics, 51(3):269–275, 1994","work_id":"6c8019b4-861e-44ba-8330-54893ef218a4","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":20,"snapshot_sha256":"532db9e794c482da344670f92430f24199d8ce5b025bdff46ad78a81a49474d0","internal_anchors":1},"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"}