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.

翻译：给定多项式 $f_0,\dots, f_k$，理想成员问题（简称 IMP）询问 $f_0$ 是否属于由 $f_1,\dots, f_k$ 生成的理想。该问题的搜索版本则要求找出这一事实的证明。IMP 是一个众所周知的基础性问题，具有众多应用。例如，它是许多基于多项式的证明系统（如零点定理、多项式演算和平方和）的基础。尽管 IMP 在一般情况下是难解的，但在许多重要情形下可以高效求解。Mastrolilli [SODA'19] 开启了针对源于约束满足问题（CSPs）的理想所对应的 IMP 的系统性研究，该问题以约束语言为参数，记为 IMP($Γ$)。这一研究方向的最终目标是根据复杂度对所有此类 IMP 进行分类。Mastrolilli 对源于 CSP($Γ$) 的 IMP 实现了这一目标，其中 $Γ$ 是布尔约束语言；而 Bulatov 和 Rafiey [STOC'22] 将这些结果推进到有限域上 CSP 的若干情形。在本文中，我们考虑源于“仿射”约束语言上 CSP 的 IMP，其中约束是阿贝尔群直积的子群（或其陪集）。这类 CSP 包括线性方程组，并被视为最重要类型的可解 CSP 之一。该问题的某些特例先前已被研究：Bharathi 和 Mastrolilli [MFCS'21] 研究了模 2 线性方程的情形，Bulatov 和 Rafiey [STOC'22] 研究了在 $GF(p)$（$p$ 为素数）上的线性方程组。本文中我们证明，若 $Γ$ 是一个仿射约束语言，则在假设输入多项式具有有界次数的前提下，IMP($Γ$) 可在多项式时间内求解。