发表机构
IIT Delhi; CNRS, IRIF; CNRS, LAMA; University of Oxford（印度理工学院德里分校; 法国国家科学研究中心，IRIF研究所; 法国国家科学研究中心，LAMA实验室; 牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将近似多项式可满足性问题（APS）的复杂性上界从PSPACE改进为计数层级（CH），并给出从希尔伯特零点定理到APS的多项式时间归约，从而更接近精确多项式可行性的复杂性。
AI 中文摘要
近似多项式可满足性问题（APS）由Guo、Saxena和Sinhababu（CCC 2018）提出，询问零向量是否位于给定多项式映射像的Zariski闭包中。具体而言，对于域$k$及其代数闭包$K$，该问题询问对于多项式映射$\boldsymbol f=(f_1,\ldots,f_m)$（其中$f_i\in k[X_1,\ldots,X_n]$），是否有$\boldsymbol 0 \in\overline{\boldsymbol f(K^n)}$。APS是希尔伯特零点定理的自然拓扑类比，即给定多项式方程组是否有公共零点的问题。APS涵盖了代数复杂性中的若干问题，包括边界秩、边界类的命中集以及零锥成员问题；已知该问题为NP难且属于PSPACE。我们证明APS在有理数和有限域上均属于计数层级（CH），显著改进了已知的PSPACE上界。我们的证明基于Andrews、Garg和Schost（FOCS 2026）关于在CH中判定希尔伯特零点定理的近期突破。作为推论，我们的结果将边界类命中集的认证复杂性从PSPACE改进为CH。我们还给出了从希尔伯特零点定理到APS的多项式时间归约，该归约适用于任意特征。在特征零的情况下，我们给出了从APS到实闭域存在性理论判定问题的归约。总体而言，我们的结果使近似多项式可满足性在复杂性上更接近精确多项式可行性，并作为副产品，为近似复杂性中出现的若干问题提供了改进的复杂性界。
英文摘要
The Approximate polynomial satisfiability problem (APS), introduced by Guo, Saxena, and Sinhababu (CCC 2018), asks whether the zero vector lies in the Zariski closure of the image of a given polynomial map. Specifically, for a field $k$ with algebraic closure~$K$, the problem asks whether $\boldsymbol 0 \in\overline{\boldsymbol f(K^n)}$ for a polynomial map $\boldsymbol f=(f_1,\ldots,f_m)$ with $f_i\in k[X_1,\ldots,X_n]$. APS is a natural topological analogue of Hilbert's Nullstellensatz, namely the question of whether a given system of polynomial equations has a common zero. APS captures several problems in algebraic complexity, including border rank, hitting sets for border classes, and null-cone membership; it is known to be NP-hard and in PSPACE. We show that APS lies in the Counting Hierarchy (CH) over both the rationals and finite fields, substantially improving the known PSPACE upper bound. Our proof builds on a recent breakthrough due to Andrews, Garg, and Schost (FOCS 2026) on deciding Hilbert's Nullstellensatz in CH. As a corollary, our result improves the complexity of certifying hitting sets for border classes from PSPACE to CH. We also give a polynomial-time reduction of Hilbert's Nullstellensatz to APS, valid in any characteristic. In characteristic zero, we give a reduction of APS to the decision problem for the existential theory of real closed fields. Overall, our results place approximate polynomial satisfiability closer in complexity to exact polynomial feasibility and as a byproduct give improved complexity bounds for several problems arising in approximative complexity.
