只读不经意ABPs子类间的精细分离
On Finer Separations between Subclasses of Read-once Oblivious ABPs
摘要：
只读不经意代数分支程序（ROABP）将多项式计算为具有矩阵作为系数的单变量多项式的乘积。在试图了解景观的代数复杂性类周围的ROABPs，我们研究类的ROABPs的基础上，这些系数矩阵的代数结构。我们研究了由这些结构化变体的ROABPs和其他著名的多项式类（如深度三供电电路，张量秩和Waring秩的多项式）计算的多项式之间的连接。我们的主要结果涉及交换ROABPs，其中所有的系数矩阵相互交换，对角ROABPs，其中所有的系数矩阵只是对角矩阵。特别是，我们展示了这些模型和与多项式的Waring秩相关的深度三供电电路模型之间的某种令人惊讶的联系。我们表明，如果维偏导数捕获Waring秩多项式的因素，那么对角ROABPs模型有效地模拟了看似更有表现力的模型交换ROABPs。此外，不能由对角ROABP有效地模拟的交换ROABP将给出显式多项式，该显式多项式给出偏导数的维数和华林秩之间的超多项式分离。我们对上述结果的证明建立在Marinari，Möller和Mora（1993）以及Möller和Stetter（1995）的结果之上，即交换矩阵的环用具有小维数偏导数的多项式表示。这些ROABPs的系数矩阵的代数结构在我们的证明中起着至关重要的作用。
英文摘要：
Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials as products of univariate polynomials that have matrices as coefficients. In an attempt to understand the landscape of algebraic complexity classes surrounding ROABPs, we study classes of ROABPs based on the algebraic structure of these coefficient matrices. We study connections between polynomials computed by these structured variants of ROABPs and other well-known classes of polynomials (such as depth-three powering circuits, tensor-rank and Waring rank of polynomials). Our main result concerns commutative ROABPs, where all coefficient matrices commute with each other, and diagonal ROABPs, where all the coefficient matrices are just diagonal matrices. In particular, we show a somewhat surprising connection between these models and the model of depth-three powering circuits that is related to the Waring rank of polynomials. We show that if the dimension of partial derivatives captures Waring rank up to polynomial factors, then the model of diagonal ROABPs efficiently simulates the seemingly more expressive model of commutative ROABPs. Further, a commutative ROABP that cannot be efficiently simulated by a diagonal ROABP will give an explicit polynomial that gives a super-polynomial separation between dimension of partial derivatives and Waring rank. Our proof of the above result builds on the results of Marinari, Möller and Mora (1993), and Möller and Stetter (1995), that characterise rings of commuting matrices in terms of polynomials that have small dimension of partial derivatives. The algebraic structure of the coefficient matrices of these ROABPs plays a crucial role in our proofs.
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