High Probability Analysis of the Condition Number of Sparse Polynomial Systems
Gregorio Malajovich (UFRJ, Rio de Janeiro, Brasil), J. Maurice Rojas (Texas A M University)
让 F:=(f_1,...,f_n) 成为一个具有固定 n 元支持的随机多项式系统。 我们的主要结果是在一个区域 U 中 f 的条件数大于 1/epsilon 的概率的上限。 边界取决于托里克歧体上一个微分形式的组成部分,当牛顿多顶(和底层协方差)都相同时,它承认一个简单的显式上界。 我们还考虑具有实系数的多项式,并为实际根数和(受限)条件数的预期数量给出界限。 使用Kahler几何框架,我们还表示区域U内f的预计根数为某种混合体积形式的U的积分,从而在U = (C^*)^n时恢复经典混合体积。
Let F:=(f_1,...,f_n) be a random polynomial system with fixed n-tuple of supports. Our main result is an upper bound on the probability that the condition number of f in a region U is larger than 1/epsilon. The bound depends on an integral of a differential form on a toric manifold and admits a simple explicit upper bound when the Newton polytopes (and underlying covariances) are all identical. We also consider polynomials with real coefficients and give bounds for the expected number of real ro...