Describing Multivariate Polynomial Subalgebras Using Equations

Abstract

Let $\mathbb{K}$ be an algebraically closed field, and $A\subset \mathbb{K}[x_1,\dots ,x_n]$ 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 $L_i:f\to c(f(\alpha )-f(\beta ))$ for some $\alpha ,\beta \in {\mathbb{K}}^n$ 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 ${\mathbb{K}}^n$.

Submitted to the special issue of Applicable Algebra in Engineering, Communication and Computing dedicated to the first conference Gröbner free methods and their applications