Describing Multivariate Polynomial Subalgebras Using Equations

TL;DR

This paper explores the structure of subalgebras of polynomial rings with finite codimension, showing that derivation-based linear functionals can be expressed as combinations of partial derivatives at points in the field.

Contribution

It characterizes the linear functionals defining filtrations of polynomial subalgebras, demonstrating derivations can be represented as linear combinations of partial derivatives evaluated at points.

Findings

Derivations in the filtration can be expressed as linear combinations of partial derivatives.

Filtrations are constructed via kernels of specific linear functionals.

Main result links derivations to evaluations of partial derivatives at points.

Abstract

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 $\mathbb{K}^{n}$.