Guo, Saxena, and Sinhababu (TOC'18, CCC'18) defined a natural, approximative analog of the polynomial system satisfiability problem, which they called approximate polynomial satisfiability (APS). They proved algebraic and geometric properties of it and showed an NP-hardness lower bound and a PSPACE upper bound for it. They further established how the problem naturally occurs in border complexity and Geometric complexity theory (GCT) and used the problem to construct hitting sets for $\overline{VP}$ in PSPACE, hence greatly mitigating the GCT chasm.

The starting point of this work is the observation that Guo, Saxena, and Sinhababu's criterion for non-existence of approximative solution can be interpreted as an analog of Weak Hilbert's Nullstellensatz in the approximative setting. We extend their work by proving an analog of Strong Hilbert's Nullstellensatz in the approximative setting. Concretely, we give an algebraic criterion for containment between approximative solution sets defined by systems of polynomials. In fact, this characterization turns out to be equivalent to membership in the integral closure over a maximal ideal of a local subring of $\mathbb{C}(x_1,\ldots,x_n)$ determined by the given polynomials. In addition, we use our proof to provide a PSPACE algorithm for testing this containment, exponentially better than the EXPSPACE bounds for polynomial subalgebra mebership testing and the polynomial integral closure membership testing, hence matching the complexity bound of Guo, Saxena, and Sinhababu's Weak Approximative Nullstellensatz.