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($\Gamma$). 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($\Gamma$) where $\Gamma$ is a Boolean constraint language, while Bulatov and Rafiey [ArXiv'21] 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 [ArXiv'21] to systems of linear equations over $GF(p)$, $p$ prime. Here we prove that if $\Gamma$ is an affine constraint language then IMP($\Gamma$) is solvable in polynomial time assuming the input polynomial has bounded degree.
翻译:鉴于多功能性 $f_0,\ dots, f_k$Imart 成员问题, IMP 简短, 询问 $f_ 0$ 是否属于 $f_ 1,\ dots, f_k$创造的理想。 在这个问题的搜索版本中, 任务是找到一个事实的证据。 IMP 是许多应用程序中众所周知的根本问题, 例如, 它基于多功能性系统, 如 Nullstellensatz, 多边性calcolulus, 和 Sum- squal 。 虽然 IMP 通常难以解决, 但在许多重要的情况下, 它可以有效解决。 Mastrolillire [SOD'19] 开始系统系统系统对于来自 Constraint Constrain Confilation 问题(CSP) 的理念进行系统系统系统系统系统系统系统研究, 也就是将所有IMPs 的功能性语言分类, 马斯特罗林性 和 IMP 最高级的 IMP 目标, 而 CSP 的 CSP 直流 直流性 直流 直径性 直径性 直径 直径性 直径 直径 直径 直径 的 直径 直 直 直 直 直 。