{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/bezoutians-and-the-mathbb-a-1-degree","title":"Bézoutians and the $\\mathbb{A}^1$-degree","arxiv_id":"2103.16614","date":"2021-03-30","proceeding":null,"authors":["Thomas Brazelton","Stephen McKean","Sabrina Pauli"],"abstract":"We prove that both the local and global $\\mathbb{A}^1$-degree of an endomorphism of affine space can be computed in terms of the multivariate B\\'ezoutian. In particular, we show that the B\\'ezoutian bilinear form, the Scheja--Storch form, and the $\\mathbb{A}^1$-degree for complete intersections are isomorphic. Our global theorem generalizes Cazanave's theorem in the univariate case, and our local theorem generalizes Kass--Wickelgren's theorem on EKL forms and the local degree. This result provides an algebraic formula for local and global degrees in motivic homotopy theory.","url_abs":"https://arxiv.org/abs/2103.16614v4","url_pdf":"https://arxiv.org/pdf/2103.16614v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"bezoutians-and-the-mathbb-a-1-degree","repo_url":"https://github.com/shmckean/A1-degree","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}