@article{40aef394-31c2-4ef6-a682-04dac681eb5c,
  abstract     = {{<p>In this paper we continue the work of describing polynomial subalgebras of finite codimension that was started in Grönkvist et al. (Appl Algebra Eng Commun Comput 33(6):751–789, 2022). Let K be an algebraically closed field, and A⊂K[x1,…,xn] be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of K-algebras (Formula presented.) where each Ai can be written as the kernel of some linear functional Li+1:Ai+1→K, and each Li is either a derivation or of the form Li:f→c(f(α)-f(β)) for some α,β∈Kn and c∈K. We investigate the structure of these filtrations and linear functionals. Our main result shows that each such Li which is a derivation may be written as a linear combination of partial derivatives evaluated at points of Kn.</p>}},
  author       = {{Leffler, Erik}},
  issn         = {{0938-1279}},
  keywords     = {{Defining conditions; Derivation; Polynomial subalgebra; Subalgebra spectrum}},
  language     = {{eng}},
  publisher    = {{Springer}},
  series       = {{Applicable Algebra in Engineering, Communications and Computing}},
  title        = {{Describing multivariate polynomial subalgebras using equations}},
  url          = {{http://dx.doi.org/10.1007/s00200-026-00743-8}},
  doi          = {{10.1007/s00200-026-00743-8}},
  year         = {{2026}},
}