Describing Multivariate Polynomial Subalgebras Using Equations

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 $$A = A_{0} \subset A_{1} \subset \ldots \subset A_m = \mathbb{K}[x_{1}, \ldots, x_n],$$ where each $A_i$ can be written as the kernel of some linear functional $L_{i + 1} : A_{i + 1} \to \mathbb{K}$, and each $L_i$ is either a derivation or of the form $L_i : f \to c(f(\mathbf{\alpha}) - f(\mathbf{\beta}))$ for some $\mathbf{\alpha}, \mathbf{\beta} \in \mathbb{K}^{n}$ 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 $\mathbb{K}^{n}$.