Computing Polynomial Representation in Subrings of Multivariate Polynomial Rings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908344017485824 |
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| author | Vu, Thi Xuan |
| author_facet | Vu, Thi Xuan |
| contents | Let $\mathcal{R} = \mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$ of characteristic 0. Consider $n$ algebraically independent elements $g_1, \dots, g_n$ in $\mathcal{R}$. Let $\mathcal{S}$ denote the subring of $\mathcal{R}$ generated by $g_1, \dots, g_n$, and let $h$ be an element of $\mathcal{S}$. Then, there exists a unique element ${f} \in \mathbb{K}[u_1, \dots, u_n]$ such that $h = f(g_1, \dots, g_n)$.
In this paper, we provide an algorithm for computing ${f}$, given $h$ and $g_1, \dots, g_n$. The complexity of our algorithm is linear in the size of the input, $h$ and $g_1, \dots, g_n$, and polynomial in $n$ when the degree of $f$ is fixed. Previous works are mostly known when $f$ is a symmetric polynomial and $g_1, \dots, g_n$ are elementary symmetric, homogeneous symmetric, or power symmetric polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computing Polynomial Representation in Subrings of Multivariate Polynomial Rings Vu, Thi Xuan Symbolic Computation Computational Complexity Algebraic Geometry Let $\mathcal{R} = \mathbb{K}[x_1, \dots, x_n]$ be a multivariate polynomial ring over a field $\mathbb{K}$ of characteristic 0. Consider $n$ algebraically independent elements $g_1, \dots, g_n$ in $\mathcal{R}$. Let $\mathcal{S}$ denote the subring of $\mathcal{R}$ generated by $g_1, \dots, g_n$, and let $h$ be an element of $\mathcal{S}$. Then, there exists a unique element ${f} \in \mathbb{K}[u_1, \dots, u_n]$ such that $h = f(g_1, \dots, g_n)$. In this paper, we provide an algorithm for computing ${f}$, given $h$ and $g_1, \dots, g_n$. The complexity of our algorithm is linear in the size of the input, $h$ and $g_1, \dots, g_n$, and polynomial in $n$ when the degree of $f$ is fixed. Previous works are mostly known when $f$ is a symmetric polynomial and $g_1, \dots, g_n$ are elementary symmetric, homogeneous symmetric, or power symmetric polynomials. |
| title | Computing Polynomial Representation in Subrings of Multivariate Polynomial Rings |
| topic | Symbolic Computation Computational Complexity Algebraic Geometry |
| url | https://arxiv.org/abs/2504.21708 |