Describing Multivariate Polynomial Subalgebras Using Equations
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917361675665408 |
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| author | Leffler, Erik |
| author_facet | Leffler, Erik |
| contents | 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 $L_i : f \to c(f(\mathbfα) - f(\mathbfβ))$ for some $\mathbfα, \mathbfβ \in \mathbb{K}^{n}$ 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 $\mathbb{K}^{n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24404 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Describing Multivariate Polynomial Subalgebras Using Equations Leffler, Erik Commutative Algebra 13P05 \and 13P10 \and 13P15 \and 12H05 13P05 \and 13P10 \and 13P15 \and 12H05 13P05 \and 13P10 \and 13P15 \and 12H05 13P05, 13P10, 13P15, 12H05 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 $L_i : f \to c(f(\mathbfα) - f(\mathbfβ))$ for some $\mathbfα, \mathbfβ \in \mathbb{K}^{n}$ 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 $\mathbb{K}^{n}$. |
| title | Describing Multivariate Polynomial Subalgebras Using Equations |
| topic | Commutative Algebra 13P05 \and 13P10 \and 13P15 \and 12H05 13P05 \and 13P10 \and 13P15 \and 12H05 13P05 \and 13P10 \and 13P15 \and 12H05 13P05, 13P10, 13P15, 12H05 |
| url | https://arxiv.org/abs/2603.24404 |