关于自由体上的恒等式检验和非交换秩计算
On Identity Testing and Noncommutative Rank Computation over the Free Skew Field
摘要：
自由斜域上有理公式的恒等式检验问题有效地归结为计算矩阵的秩，矩阵的元素是关于非交换变量\cite{HW15}的线性多项式。该秩计算问题具有确定性多项式时间白盒算法\cite{GGOW16, IQS18}和黑盒设置中的随机多项式时间算法\cite{DM17}。在本文中，我们提出了一种新的方法，有效的去随机化的\emph{black-box} RIT。此外，我们还得到了自由斜域上矩阵秩的计算结果，并为一类新的有理表达式构造了有效的线性束表示。更准确地说，我们显示了以下结果：1。在严格假设$k\times k$矩阵代数的每个多项式恒等式的ABP（代数分支程序）复杂性为$2^{Ω(k)}$ \cite{BW05}的条件下，得到了几乎一般情况下RIT的一个次指数时间黑箱算法.这可以被看作是第一个有理公式的“硬度隐含去随机化”型定理。2.我们证明了在自由体上任何矩阵的非交换秩都可以在确定性多项式时间内计算，该矩阵的元素具有小的线性束表示。在此之前，一个有效的秩计算只知道与非交换公式作为条目\cite{GGOW20}的矩阵。作为我们算法的特殊情况，我们得到了第一个确定性的多项式时间算法的秩计算矩阵的条目是非交换ABPs或有理公式。3.受Bergman \cite{Ber76}定义的启发，我们定义了一个包含非交换ABPs和有理公式的新类.我们得到了这个类的多项式大小的线性铅笔表示。作为一个副产品，我们得到了一个白盒确定性的多项式时间的身份测试算法的类。
英文摘要：
The identity testing of rational formulas (RIT) in the free skew field efficiently reduces to computing the rank of a matrix whose entries are linear polynomials in noncommuting variables\cite{HW15}. This rank computation problem has deterministic polynomial-time white-box algorithms \cite{GGOW16, IQS18} and a randomized polynomial-time algorithm in the black-box setting \cite{DM17}. In this paper, we propose a new approach for efficient derandomization of \emph{black-box} RIT. Additionally, we obtain results for matrix rank computation over the free skew field, and construct efficient linear pencil representations for a new class of rational expressions. More precisely, we show the following results: 1. Under the hardness assumption that the ABP (algebraic branching program) complexity of every polynomial identity for the $k\times k$ matrix algebra is $2^{Ω(k)}$ \cite{BW05}, we obtain a subexponential-time black-box algorithm for RIT in almost general setting. This can be seen as the first "hardness implies derandomization" type theorem for rational formulas. 2. We show that the noncommutative rank of any matrix over the free skew field whose entries have small linear pencil representations can be computed in deterministic polynomial time. Prior to this, an efficient rank computation was only known for matrices with noncommutative formulas as entries\cite{GGOW20}. As special cases of our algorithm, we obtain the first deterministic polynomial-time algorithms for rank computation of matrices whose entries are noncommutative ABPs or rational formulas. 3. Motivated by the definition given by Bergman\cite{Ber76}, we define a new class that contains noncommutative ABPs and rational formulas. We obtain a polynomial-size linear pencil representation for this class. As a by-product, we obtain a white-box deterministic polynomial-time identity testing algorithm for the class.
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