代数依赖与近似复杂度下的PSPACE算法
Algebraic dependencies and PSPACE algorithms in approximative complexity
摘要：
测试一组$\mathbf{f}$多项式是否具有代数依赖性是一个具有几个应用的基本问题。多项式作为代数电路给出。代数独立性测试问题在有限域上是广泛开放的（Dvir，Gabizon，Wigderson，FOCS'07）。已知的最好的复杂性是NP $^{\#\rm P}$（Mittmann，Saxena，Scheiblechner，Trans.AMS'14）。在这项工作中，我们把问题的AM $\cap$ coAM。特别是，依赖测试不太可能是NP难的，并加入了“中等”复杂性问题的联盟，例如。图同构与整数分解我们的证明方法是代数几何-估计有限域上多项式映射$\mathbf{f}$的像/原像的大小。在AM协议中利用这种大小的间隙。接下来，我们研究了检验$\mathbf{f}$的每个零化子是否都有零常数项的公开问题（Kayal，CCC'09）。我们给出了一个几何特征，使用Zapriki封闭的图像的$\mathbf{f}$，介绍了一个新的问题，称为近似多项式可满足性（APS）。我们表明，APS是NP-困难的，并使用投影代数几何的思想，我们把APS在PSPACE（事先最好是EXPSPACE通过Grobner基础计算）。作为这一点在近似复杂性理论中的一个意外应用，我们得到--在任何域上，$\overline{\rm VP}$的命中集都可以在PSPACE中设计。这解决了（Mulmuley，FOCS'12，J.AMS 2017）中提出的一个开放问题;极大地缓解了GCT鸿沟（就空间复杂度而言呈指数级）。
英文摘要：
Testing whether a set $\mathbf{f}$ of polynomials has an algebraic dependence is a basic problem with several applications. The polynomials are given as algebraic circuits. Algebraic independence testing question is wide open over finite fields (Dvir, Gabizon, Wigderson, FOCS'07). The best complexity known is NP$^{\#\rm P}$ (Mittmann, Saxena, Scheiblechner, Trans.AMS'14). In this work we put the problem in AM $\cap$ coAM. In particular, dependence testing is unlikely to be NP-hard and joins the league of problems of "intermediate" complexity, eg. graph isomorphism & integer factoring. Our proof method is algebro-geometric-- estimating the size of the image/preimage of the polynomial map $\mathbf{f}$ over the finite field. A gap in this size is utilized in the AM protocols. Next, we study the open question of testing whether every annihilator of $\mathbf{f}$ has zero constant term (Kayal, CCC'09). We give a geometric characterization using Zariski closure of the image of $\mathbf{f}$; introducing a new problem called approximate polynomials satisfiability (APS). We show that APS is NP-hard and, using projective algebraic-geometry ideas, we put APS in PSPACE (prior best was EXPSPACE via Grobner basis computation). As an unexpected application of this to approximative complexity theory we get-- Over any field, hitting-set for $\overline{\rm VP}$ can be designed in PSPACE. This solves an open problem posed in (Mulmuley, FOCS'12, J.AMS 2017); greatly mitigating the GCT Chasm (exponentially in terms of space complexity).
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