We show that Hilbert's Nullstellensatz, the problem of deciding if a system of multivariate polynomial equations has a solution in the algebraic closure of the underlying field, lies in the counting hierarchy. More generally, we show that the number of solutions to a system of equations can be computed in polynomial time with oracle access to the counting hierarchy. Our results hold in particular for polynomials with coefficients in either the rational numbers or a finite field. Previously, the best-known bounds on the complexities of these problems were PSPACE and FPSPACE, respectively. Our main technical contribution is the construction of a uniform family of constant-depth arithmetic circuits that compute the multivariate resultant.

翻译：我们证明希尔伯特零点定理——即判定多元多项式方程组在基域的代数闭包中是否存在解的问题——属于计数层级。更一般地，我们证明方程组的解的数量可以通过以计数层级为谕示的多项式时间算法计算。我们的结果特别适用于系数为有理数或有限域的多项式。此前，这些问题已知的最佳复杂度上界分别为PSPACE和FPSPACE。我们的主要技术贡献是构建了一个计算多元结式的恒定深度算术电路的一致族。