The Ideal Membership Problem and Abelian Groups
Andrei A. Bulatov, Akbar Rafiey
TL;DR
This work studies the Ideal Membership Problem (IMP) through the lens of Constraint Satisfaction Problems (CSPs), focusing on affine constraint languages over finite Abelian groups. It develops a comprehensive multi-sorted framework, pp-definability, and pp-interpretations to reduce IMPs to systems of linear equations over cyclic $p$-groups, which are then transformed to root-of-unity formulations and solved via Gröbner bases. The main contributions show that if the constraint language Γ is invariant under the affine operation x−y+z on a finite Abelian group, then IMP_d(Γ) is solvable in polynomial time for every degree bound d, and a degree-d Gröbner basis for the corresponding ideal can be constructed in polynomial time. The paper also introduces χIMP and a substitution-based reduction technique, enabling broad transfer of tractability to a wide class of IMP instances, including multi-sorted and Abelian-group settings, with implications for proof systems and algebraic CSPs.
Abstract
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.