We study the algebraic complexity of annihilators of polynomials maps. In particular, when a polynomial map is `encoded by' a small algebraic circuit, we show that the coefficients of an annihilator of the map can be computed in PSPACE. Even when the underlying field is that of reals or complex numbers, an analogous statement is true. We achieve this by using the class VPSPACE that coincides with computability of coefficients in PSPACE, over integers. As a consequence, we derive the following two conditional results. First, we show that a VP-explicit hitting set generator for all of VP would separate either VP from VNP, or non-uniform P from PSPACE. Second, in relation to algebraic natural proofs, we show that proving an algebraic natural proofs barrier would imply either VP $\neq$ VNP or DSPACE($\log^{\log^{\ast}n} n$) $\not\subset$ P.

翻译：本文研究多项式映射零化子的代数复杂度。特别地，当多项式映射由小型代数电路"编码"时，我们证明该映射零化子的系数可在PSPACE中计算。即使底层域为实数域或复数域，相应的结论仍然成立。我们通过利用与整数域上PSPACE中系数可计算性相一致的VPSPACE类来实现这一结果。作为推论，我们得到以下两个条件性结论：首先，我们证明针对整个VP类的VP-显式击中集生成元要么将VP与VNP分离，要么将非均匀P与PSPACE分离；其次，在与代数自然证明的关系中，我们证明建立代数自然证明屏障将蕴含要么VP ≠ VNP，要么DSPACE($\log^{\log^{\ast}n} n$) ⊄ P。