Suppose $f,g_1,\cdots,g_p$ are holomorphic functions over $\Omega\subset\cxC^n$. Then there raises a natural question: when can we find holomorphic functions $h_1,\cdots,h_p$ such that $f=\sum g_jh_j$? The celebrated Skoda theorem solves this question and gives a $L^2$ sufficient condition. In general, we can consider the vector bundle case, i.e. to determine the sufficient condition of solving $f_i(x)=\sum g_{ij}(x)h_j(x)$ with parameter $x\in\Omega$. Since the problem is related to solving linear equations, the answer naturally connects to the Cramer's rule. In the first part we will give a proof of division theorem by projectivization technique and study the generalized fundamental inequalities. In the second part we will apply the skills and the results of the division theorems to show some applications. / Mathematics

Systems of linear equations are central to many science and engineering application domains. Given the abundance of low-cost parallel processing fabrics, the study of fast and accurate parallel algorithms for solving such systems is receiving attention. Fast linear solvers generally use a form of LU factorization. These methods face challenges with workload distribution and communication overhead that hinder their application in a true broadcast communication environment. Presented is an efficient framework for solving large-scale linear systems by means of a novel utilization of Cramer's rule. While the latter is often perceived to be impractical when considered for large systems, it is shown that the algorithm proposed has an order N^3 complexity with pragmatic forward and backward stability. To the best of our knowledge, this is the first time that Cramer's rule has been demonstrated to be an order N^3 process. Empirical results are provided to substantiate the stated accuracy and computational complexity, clearly demonstrating the efficacy of the approach taken. The unique utilization of Cramer's rule and matrix condensation techniques yield an elegant process that can be applied to parallel computing architectures that support a broadcast communication infrastructure. The regularity of the communication patterns, and send-ahead ability, yields a viable framework for solving linear equations using conventional computing platforms. In addition, this dissertation demonstrates the algorithm's potential for solving large-scale sparse linear systems.