complexity theory teams

Abstract: Andrews, Garg, and Schost 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.

Abstract: The Aharonov-Regev proof that $\mathrm{GapCVP}_{c\sqrt n}$ lies in $\mathsf{NP}\cap \mathsf{coNP}$ uses a verifier that tests dual lattice vectors; its proof gives $c=100$. We study this verifier through a linear program in which a certificate is modeled as a sample from a probability distribution on the dual lattice. This viewpoint gives an exact analysis of the close-target tests: adding gradient and Hessian checks increases the certified close radius from $σ^{-1}/4$ to $σ^{-1}\sqrt{3/16}$, and higher derivatives cannot improve it within this framework. With Gaussian certificates for far targets, we obtain $$\mathrm{GapCVP}_{c\sqrt n},\ \mathrm{GapSVP}_{c\sqrt n}\in \mathsf{NP} \cap\mathsf{coNP}$$ for every $c>2/(π\sqrt3)\approx0.3676$. An appendix gives a finite-bit implementation of the verifier. We also show that the Gaussian certificate is not always optimal and solve the unbounded-support program exactly for targets of order two, where its value is determined by the spectral capacity of an odd dual coset.