🤖 AI Summary
This paper addresses the explicit representation problem for subrings of multivariate polynomial rings over fields of characteristic zero: given algebraically independent generators $g_1,dots,g_n$ and an element $h in K[g_1,dots,g_n]$, compute the unique polynomial $f in K[u_1,dots,u_n]$ satisfying $h = f(g_1,dots,g_n)$. We propose the first general-purpose algorithm applicable to arbitrary algebraically independent generators—surpassing prior approaches restricted to symmetric polynomials. Our method integrates Gröbner basis techniques, variable substitution, and sparse interpolation, augmented by degree bounds and sparsity-aware optimizations. The algorithm runs in time linear in the input size and polynomial in $n$ (for fixed $deg f$), and we provide a rigorous correctness proof. Experimental evaluation confirms its efficiency and practicality on non-symmetric generator sets, demonstrating substantial speedups over existing methods.
📝 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.