@article{18617,
  abstract     = {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.
Formally, 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.},
  author       = {Beďatš, Daniel},
  issn         = {0021-8693},
  journal      = {Journal of Algebra},
  pages        = {281--304},
  publisher    = {Elsevier},
  title        = {{Separation of variables for scalar-valued polynomials in the non-stable range}},
  doi          = {10.1016/j.jalgebra.2024.04.013},
  volume       = {651},
  year         = {2024},
}