代数独立性与黑盒恒等式检验
Algebraic Independence and Blackbox Identity Testing
摘要：
代数独立性是交换代数中的一个高级概念，它将线性多项式的独立性推广到更高的次数。多项式{f_1，.，f_m} \subset \F [x_1，.，x_n]称为代数无关的，如果不存在非零多项式F使得F（f_1，...，f_m）= 0。超越度，trdeg{f_1，.，f_m}是集合中代数无关多项式的最大个数r。在本文中，我们设计了黑箱和有效的线性映射，使变量的数量从n减少到r，但保持trdeg{f（f_i）}_i = r，假设f_i的稀疏和小的r。我们应用这些基本映射来解决黑盒身份测试的几种情况：（1）给定多项式次数电路C和稀疏多项式f_1，.，f_m与trdeg r，我们可以测试黑盒D：= C（f_1，...，f_m）在poly（size（D））中为零。(2)将spsp_δ（k，s，n）电路C定义为\sum_{i=1}^k \prod_{j=1}^s f_{i,j}的形式，其中f_{i,j}是次数至多为δ的稀疏n变量多项式。对于k = 2，我们给出了一个poly（snδ）^{δ^2}时间黑盒恒等式检验. (3)对于一般的深度4电路，我们定义了秩的概念。假设最小简单spsp_δ（k，s，n）恒等式存在秩界R，给出了spsp_δ（k，s，n）电路的poly（snRδ）^{Rkδ^2}时间黑盒恒等式检验.这部分地概括了深度3至深度4电路的现有技术。trdeg的概念最适合大或零特征，但我们也给出了我们的结果的版本为任意字段。
英文摘要：
Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. Polynomials {f_1, ..., f_m} \subset \F[x_1, ..., x_n] are called algebraically independent if there is no non-zero polynomial F such that F(f_1, ..., f_m) = 0. The transcendence degree, trdeg{f_1, ..., f_m}, is the maximal number r of algebraically independent polynomials in the set. In this paper we design blackbox and efficient linear maps ϕthat reduce the number of variables from n to r but maintain trdeg{ϕ(f_i)}_i = r, assuming f_i's sparse and small r. We apply these fundamental maps to solve several cases of blackbox identity testing: (1) Given a polynomial-degree circuit C and sparse polynomials f_1, ..., f_m with trdeg r, we can test blackbox D := C(f_1, ..., f_m) for zeroness in poly(size(D))^r time. (2) Define a spsp_δ(k,s,n) circuit C to be of the form \sum_{i=1}^k \prod_{j=1}^s f_{i,j}, where f_{i,j} are sparse n-variate polynomials of degree at most δ. For k = 2 we give a poly(snδ)^{δ^2} time blackbox identity test. (3) For a general depth-4 circuit we define a notion of rank. Assuming there is a rank bound R for minimal simple spsp_δ(k,s,n) identities, we give a poly(snRδ)^{Rkδ^2} time blackbox identity test for spsp_δ(k,s,n) circuits. This partially generalizes the state of the art of depth-3 to depth-4 circuits. The notion of trdeg works best with large or zero characteristic, but we also give versions of our results for arbitrary fields.
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