Computing Polynomial Representation in Subrings of Multivariate Polynomial Rings
Thi Xuan Vu
TL;DR
The paper studies representing a polynomial $h$ in a subring $\\mathcal{S}=\\mathbb{K}[g_1,\\dots, g_n]$ of a multivariate polynomial ring, showing that there exists a unique $f\\in\\mathbb{K}[u_1,\\dots, u_n]$ with $h=f(g_1,\\dots, g_n)$. It introduces a randomized Representation algorithm based on Newton–Hensel lifting to compute $f$ from $h$ and the $g_i$, with complexity $\\tilde{O}((nL_1+n^4+L_2)\\mathcal{M}(\\Delta, n))$ where $L_1,L_2$ are the input SLP lengths and $\\Delta$ bounds the degree of $f$. The method extends to cases where $h$ is invariant under finite pseudo-reflection groups, yielding $\\Delta\\le\\deg(h)$ and complexity tied to $\\mathcal{M}(\\deg(h),n)$. The paper also establishes degree-bounds via weighted degrees, discusses invariant theory implications, and outlines directions for decomposing multivariate polynomials and further improving invariant-system computations.
Abstract
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.