{"paper":{"title":"Graphical Algebraic Geometry: From Ideals and Varieties to Quantum Calculi","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Graphical Algebraic Geometry supplies diagrammatic languages that are universal and complete for commutative algebras and affine varieties.","cross_cats":["cs.LO","math.CT"],"primary_cat":"quant-ph","authors_text":"Aleks Kissinger, Dichuan Gao, Razin A. Shaikh","submitted_at":"2026-05-13T18:05:02Z","abstract_excerpt":"We introduce Graphical Algebraic Geometry (GAG), a family of diagrammatic languages extending the Graphical Linear Algebra programme. We construct several languages within this family and prove that they are universal and complete for the corresponding (co)span semantics of commutative algebras and affine varieties. This framework provides clear graphical representations of algebraic structures -- such as polynomials, ideals, and varieties -- enabling intuitive yet rigorous diagrammatic reasoning. We showcase two practical viewpoints on GAG. First, we show that instances of counting constraint"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We construct several languages within this family and prove that they are universal and complete for the corresponding (co)span semantics of commutative algebras and affine varieties.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that the chosen diagrammatic generators and relations exactly capture the (co)span semantics of commutative algebras without introducing or losing algebraic information.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Graphical Algebraic Geometry creates universal diagrammatic languages for commutative algebras and affine varieties that also characterize the qudit ZH calculus for quantum computation.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Graphical Algebraic Geometry supplies diagrammatic languages that are universal and complete for commutative algebras and affine varieties.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a318262745fb59c30c8d735ce31b5ce54083019f5e16269400f91a8b04e7c85f"},"source":{"id":"2605.13993","kind":"arxiv","version":1},"verdict":{"id":"ec1751a2-c32c-4d1a-9902-32b8c2dad02d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T05:40:39.250453Z","strongest_claim":"We construct several languages within this family and prove that they are universal and complete for the corresponding (co)span semantics of commutative algebras and affine varieties.","one_line_summary":"Graphical Algebraic Geometry creates universal diagrammatic languages for commutative algebras and affine varieties that also characterize the qudit ZH calculus for quantum computation.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that the chosen diagrammatic generators and relations exactly capture the (co)span semantics of commutative algebras without introducing or losing algebraic information.","pith_extraction_headline":"Graphical Algebraic Geometry supplies diagrammatic languages that are universal and complete for commutative algebras and affine varieties."},"references":{"count":75,"sample":[{"doi":"","year":2017,"title":"Geometry of 3D Environments and Sum of Squares Polynomials","work_id":"ce7f08b1-ab4e-41c2-bdb5-ca0451b2cc6a","ref_index":1,"cited_arxiv_id":"1611.07369","is_internal_anchor":true},{"doi":"10.1103/physreva","year":2001,"title":"Asymptotically Good Quantum Codes","work_id":"bbc98446-d3ae-46ab-8856-14aca325ac62","ref_index":2,"cited_arxiv_id":"quant-ph/0006061","is_internal_anchor":true},{"doi":"10.4204/eptcs","year":2019,"title":"Miriam Backens and Aleks Kissinger. 2019. 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