理想成员问题与阿贝尔群
The Ideal Membership Problem and Abelian Groups
AI总结：
本文证明，对于仿射约束语言，理想成员问题可以在多项式时间内解决，前提是输入多项式次数有界。
摘要：
给定多项式$f_0,\dots, f_k$，理想成员问题（IMP）询问$f_0$是否属于由$f_1,\dots, f_k$生成的理想。在搜索版本中，任务是找到证明这一事实的证明。IMP是一个广为人知的基本问题，有众多应用。例如，它支撑了许多基于多项式的证明系统，如消元定理、多项式演算和平方和。尽管IMP在一般情况下是难以处理的，但在许多重要的情况下可以高效解决。Mastrolilli [SODA'19] 开始了对来自约束满足问题（CSPs）的理想IMP的系统研究，参数化于约束语言，称为IMP($Γ$)。这一研究线的最终目标是根据其复杂性对所有此类IMP进行分类。Mastrolilli实现了这一目标，对于CSP($Γ)）中的IMP，其中Γ是布尔约束语言。Bulatov和Rafiey [STOC'22] 将这些结果扩展到CSPs的几个案例。在本文中，我们考虑来自CSPs的IMP，其中约束是阿贝尔群的直接积的子群（或其陪集）。这种CSP包括线性方程组，并被认为是最重要的可处理CSPs之一。一些特殊案例之前已被Bharathi和Mastrolilli [MFCS'21] 考虑过，针对模2的线性方程，以及Bulatov和Rafiey [STOC'22] 对GF(p)上的线性方程组。这里我们证明，如果Γ是仿射约束语言，则在输入多项式具有有界次数的前提下，IMP($Γ$)可以在多项式时间内解决。
英文摘要：
Given polynomials $f_0,\dots, f_k$ the Ideal Membership Problem, IMP for short, asks if $f_0$ belongs to the ideal generated by $f_1,\dots, f_k$. In the search version of this problem the task is to find a proof of this fact. The IMP is a well-known fundamental problem with numerous applications. For instance, it underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. Although the IMP is in general intractable, in many important cases it can be efficiently solved. Mastrolilli [SODA'19] initiated a systematic study of IMPs for ideals arising from Constraint Satisfaction Problems (CSPs), parameterized by constraint languages, denoted IMP($Γ$). The ultimate goal of this line of research is to classify all such IMPs accordingly to their complexity. Mastrolilli achieved this goal for IMPs arising from CSP($Γ$) where $Γ$ is a Boolean constraint language, while Bulatov and Rafiey [STOC'22] advanced these results to several cases of CSPs over finite domains. In this paper we consider IMPs arising from CSPs over `affine' constraint languages, in which constraints are subgroups (or their cosets) of direct products of Abelian groups. This kind of CSPs include systems of linear equations and are considered one of the most important types of tractable CSPs. Some special cases of the problem have been considered before by Bharathi and Mastrolilli [MFCS'21] for linear equation modulo 2, and by Bulatov and Rafiey [STOC'22] to systems of linear equations over $GF(p)$, $p$ prime. Here we prove that if $Γ$ is an affine constraint language then IMP($Γ$) is solvable in polynomial time assuming the input polynomial has bounded degree.
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