Describing multivariate polynomial subalgebras using equations

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 K be an algebraically closed field, and A ⊂ K[x₁, … , xₙ] A ⊂ K [ x 1 , … , x n ] be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of K K -algebras A = A₀ ⊂ A₁ ⊂ ⋯ ⊂ Aₘ = K[x₁, … , xₙ], A = A 0 ⊂ A 1 ⊂ ⋯ ⊂ A m = K [ x 1 , … , x n ] , where each Aᵢ A i can be written as the kernel of some linear functional Li + 1: Ai + 1 → K L i + 1 : A i + 1 → K , and each Lᵢ L i is either a derivation or of the form Lᵢ: f → c(f(α ) - f(β )) L i : f → c ( f ( α ) - f ( β ) ) for some α , β ∈ Kⁿ α , β ∈ K n </mml:ms