论文标题
难以检测单变量整数多项式的因子
Hard to Detect Factors of Univariate Integer Polynomials
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
We investigate the computational complexity of deciding whether a given univariate integer polynomial p(x) has a factor q(x) satisfying specific additional constraints. When the only constraint imposed on q(x) is to have a degree smaller than the degree of p(x) and greater than zero, the problem is equivalent to testing the irreducibility of p(x) and then it is solvable in polynomial time. We prove that deciding whether a given monic univariate integer polynomial has factors satisfying additional properties may lead to NP-complete problems in the strong sense. In particular, given any constant value k in Z, we prove that it is NP-complete in the strong sense to detect the existence of a factor that returns a prescribed value when evaluated at x=k or to detect the existence of a pair of factors - whose product is equal to the original polynomial - that return the same value when evaluated at x=k.
