Approximate Polynomial Satisfiability and Integral Closure Membership in the Counting Hierarchy
Andrews, Garg, and Schost (FOCS~2026) recently, in a breakthrough result, placed Hilbert's Nullstellensatz (HN) in the Counting Hierarchy ($\CH$) using constant-depth circuits for multivariate resultants. While HN deals with exact polynomial satisfiability, Guo, Saxena, and Sinhababu (CCC~2018; Theory of Computing~2019) introduced approximate polynomial satisfiability (APS) as a natural approximative analogue of polynomial system satisfiability. They proved its $\mathsf{NP}$-hardness and a $\PSPACE$ upper bound, and used APS to construct hitting sets for $\overline{\mathsf{VP}}$ in polynomial space, substantially narrowing the GCT chasm in geometric complexity theory. We place APS in the Counting Hierarchy for rational polynomials given by arithmetic circuits. Our proof combines Noether normalization with a geometric restriction on how points in the image of a polynomial map approach the origin, reducing APS to a test on multivariate resultants which allows solvability in $\CH$. It complements the independent and concurrent proofs of Balaji, Shirmohammadi, Tavenas, and Worrell, and Andrews, Garg, Oliveira, and Schost. Moreover, we also place integral closure membership testing in the Counting Hierarchy. Here, the question is whether a given polynomial $f$ is integral over $\C[f_1,\ldots,f_m]$. Our approach uses a global annihilator criterion and an effective degree bound to reduce non-integrality to APS in polynomial time. In fact, we show that the reduction holds both ways, and Integral Closure Membership Testing is equivalent to APS under polynomial-time reductions.