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 \(\mathbb {K}\) be an algebraically closed field, and \(A \subset \mathbb {K}[x_{1}, \ldots , x_n]\) be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of \(\mathbb {K}\) -algebras \( A = A_{0} \subset A_{1} \subset \cdots \subset A_m = \mathbb {K}[x_{1}, \ldots , x_n], \) where each \(A_i\) can be written as the kernel of some linear functional \(L_{i + 1}: A_{i + 1} \rightarrow \mathbb {K}\) , and each \(L_i\) is either a derivation or of the form \(L_i: f \rightarrow c(f(\varvec{\alpha }) - f(\varvec{\beta }))\) for some \(\varvec{\alpha }, \varvec{\beta }\in \mathbb {K}^{n}\) and \(c \in \mathbb {K}\) . We investigate the structure of these filtrations and linear functionals. Our main result shows that each such \(L_i\) which is a derivation may be written as a linear combination of partial derivatives evaluated at points of \(\mathbb {K}^{n}\) .
