[{"intvolume":"       651","scopus_import":"1","doi":"10.1016/j.jalgebra.2024.04.013","date_created":"2024-12-04T07:58:45Z","volume":651,"author":[{"orcid":"0009-0004-1828-0044","last_name":"Beďatš","full_name":"Beďatš, Daniel","first_name":"Daniel","id":"78ea3cc9-31e7-11ee-aa02-a6169bbfe1f1"}],"month":"08","oa":1,"status":"public","oa_version":"Published Version","_id":"18617","file_date_updated":"2024-12-09T13:56:26Z","ddc":["510"],"language":[{"iso":"eng"}],"year":"2024","OA_type":"hybrid","page":"281-304","license":"https://creativecommons.org/licenses/by/4.0/","day":"01","article_type":"original","publisher":"Elsevier","department":[{"_id":"UlWa"}],"has_accepted_license":"1","article_processing_charge":"Yes (via OA deal)","isi":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"date_published":"2024-08-01T00:00:00Z","external_id":{"isi":["001232775600001"],"arxiv":["2309.11154"]},"date_updated":"2025-09-08T14:57:00Z","publication_status":"published","title":"Separation of variables for scalar-valued polynomials in the non-stable range","quality_controlled":"1","abstract":[{"text":"Any complex-valued polynomial on (Rn)k decomposes into an algebraic combination of O(n)-invariant polynomials and harmonic polynomials. This decomposition, separation of variables, is granted to be unique if n≥2k−1. We prove that the condition n≥2k−1 is not only sufficient, but also necessary for uniqueness of the separation. Moreover, we describe the structure of non-uniqueness of the separation in the boundary cases when n=2k−2 and n=2k−3.\r\nFormally, we study the kernel of a multiplication map ϕ carrying out separation of variables. We devise a general algorithmic procedure for describing Ker ϕ in the restricted non-stable range k≤n<2k−1. In the full non-stable range n<2k−1, we give formulas for highest weights of generators of the kernel as well as formulas for its Hilbert series. Using the developed methods, we obtain a list of highest weight vectors generating Ker ϕ.","lang":"eng"}],"publication_identifier":{"issn":["0021-8693"]},"corr_author":"1","arxiv":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"ista":"Beďatš D. 2024. Separation of variables for scalar-valued polynomials in the non-stable range. Journal of Algebra. 651, 281–304.","mla":"Beďatš, Daniel. “Separation of Variables for Scalar-Valued Polynomials in the Non-Stable Range.” <i>Journal of Algebra</i>, vol. 651, Elsevier, 2024, pp. 281–304, doi:<a href=\"https://doi.org/10.1016/j.jalgebra.2024.04.013\">10.1016/j.jalgebra.2024.04.013</a>.","ieee":"D. Beďatš, “Separation of variables for scalar-valued polynomials in the non-stable range,” <i>Journal of Algebra</i>, vol. 651. Elsevier, pp. 281–304, 2024.","ama":"Beďatš D. Separation of variables for scalar-valued polynomials in the non-stable range. <i>Journal of Algebra</i>. 2024;651:281-304. doi:<a href=\"https://doi.org/10.1016/j.jalgebra.2024.04.013\">10.1016/j.jalgebra.2024.04.013</a>","short":"D. Beďatš, Journal of Algebra 651 (2024) 281–304.","chicago":"Beďatš, Daniel. “Separation of Variables for Scalar-Valued Polynomials in the Non-Stable Range.” <i>Journal of Algebra</i>. Elsevier, 2024. <a href=\"https://doi.org/10.1016/j.jalgebra.2024.04.013\">https://doi.org/10.1016/j.jalgebra.2024.04.013</a>.","apa":"Beďatš, D. (2024). Separation of variables for scalar-valued polynomials in the non-stable range. <i>Journal of Algebra</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jalgebra.2024.04.013\">https://doi.org/10.1016/j.jalgebra.2024.04.013</a>"},"publication":"Journal of Algebra","file":[{"file_id":"18638","file_size":486969,"content_type":"application/pdf","relation":"main_file","checksum":"7b01c89128ba16d5334dfab389a03878","date_created":"2024-12-09T13:56:26Z","file_name":"2024_JourAlgebra_Bedats.pdf","success":1,"date_updated":"2024-12-09T13:56:26Z","access_level":"open_access","creator":"dernst"}],"acknowledgement":"The author is sincerely grateful for guidance, advice and valuable feedback from Roman Lávička.","type":"journal_article","OA_place":"publisher"}]