---
OA_place: publisher
OA_type: hybrid
_id: '18617'
abstract:
- lang: eng
  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."
acknowledgement: The author is sincerely grateful for guidance, advice and valuable
  feedback from Roman Lávička.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Daniel
  full_name: Beďatš, Daniel
  id: 78ea3cc9-31e7-11ee-aa02-a6169bbfe1f1
  last_name: Beďatš
  orcid: 0009-0004-1828-0044
citation:
  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>
  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>
  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>.
  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.
  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>.
  short: D. Beďatš, Journal of Algebra 651 (2024) 281–304.
corr_author: '1'
date_created: 2024-12-04T07:58:45Z
date_published: 2024-08-01T00:00:00Z
date_updated: 2025-09-08T14:57:00Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1016/j.jalgebra.2024.04.013
external_id:
  arxiv:
  - '2309.11154'
  isi:
  - '001232775600001'
file:
- access_level: open_access
  checksum: 7b01c89128ba16d5334dfab389a03878
  content_type: application/pdf
  creator: dernst
  date_created: 2024-12-09T13:56:26Z
  date_updated: 2024-12-09T13:56:26Z
  file_id: '18638'
  file_name: 2024_JourAlgebra_Bedats.pdf
  file_size: 486969
  relation: main_file
  success: 1
file_date_updated: 2024-12-09T13:56:26Z
has_accepted_license: '1'
intvolume: '       651'
isi: 1
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '08'
oa: 1
oa_version: Published Version
page: 281-304
publication: Journal of Algebra
publication_identifier:
  issn:
  - 0021-8693
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Separation of variables for scalar-valued polynomials in the non-stable range
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 651
year: '2024'
...