双线性映射有界系数复杂度的下界
Lower Bounds on the Bounded Coefficient Complexity of Bilinear Maps
摘要：
在复数域上有界系数算术电路模型下，证明了$n$次多项式相乘问题和多项式带余数除法问题的阶数$n\log n$的下界.这些下限是最佳的数量级。证明使用了R的一个最近的想法。Raz [Proc.34th STOC 2002]提出了矩阵乘法。它将随机循环矩阵与向量相乘的线性问题归结为循环卷积的双线性问题。我们通过扩展J. Morgenstern的界[J. ACM 20，pp. 305-306，1973]在一个单一的不变的方式。这建立了一个新的下界的有界系数的复杂性的线性形式的奇异值的相应矩阵。此外，我们扩展这些下界的线性和双线性映射的电路模型，允许有限数量的无界标量乘法。
英文摘要：
We prove lower bounds of order $n\log n$ for both the problem to multiply polynomials of degree $n$, and to divide polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the linear problem to multiply a random circulant matrix with a vector to the bilinear problem of cyclic convolution. We treat the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp. 305-306, 1973] in a unitarily invariant way. This establishes a new lower bound on the bounded coefficient complexity of linear forms in terms of the singular values of the corresponding matrix. In addition, we extend these lower bounds for linear and bilinear maps to a model of circuits that allows a restricted number of unbounded scalar multiplications.
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