Polynomial factorization in conventional sense is an ill-posed problem due to its discontinuity with respect to coefficient perturbations, making it a challenge for numerical computation using empirical data. As a regularization, this paper formulates the notion of numerical factorization based on the geometry of polynomial spaces and the stratification of factorization manifolds. Furthermore, this paper establishes the existence, uniqueness, Lipschitz continuity, condition number, and convergence of the numerical factorization to the underlying exact factorization, leading to a robust and efficient algorithm with a MATLAB implementation capable of accurate polynomial factorizations using floating point arithmetic even if the coefficients are perturbed.

翻译：常规意义上的多元系数化是一个错误的问题,因为它在系数扰动方面的不连续性,使得它成为使用实证数据进行数字计算的一个挑战。作为正规化,本文件根据多元空间的几何学和乘数分层来拟订数字系数化概念。此外,本文件确定了数字系数化的存在、独特性、利普西茨连续性、条件号以及数字系数化与基本精确系数化的趋同,导致一种稳健有效的算法,其MATLAB实施过程能够使用浮点算术进行精确的多数值化。