In exploratory factor analysis, model parameters are usually estimated by maximum likelihood method. The maximum likelihood estimate is obtained by solving a complicated multivariate algebraic equation. Since the solution to the equation is usually intractable, it is typically computed with continuous optimization methods, such as Newton-Raphson methods. With this procedure, however, the solution is inevitably dependent on the estimation algorithm and initial value since the log-likelihood function is highly non-concave. Particularly, the estimates of unique variances can result in zero or negative, referred to as improper solutions; in this case, the maximum likelihood estimate can be severely unstable. To delve into the issue of the instability of the maximum likelihood estimate, we compute exact solutions to the multivariate algebraic equation by using algebraic computations. We provide a computationally efficient algorithm based on the algebraic computations specifically optimized for maximum likelihood factor analysis. To be specific, Gr\"oebner basis and cylindrical decomposition are employed, powerful tools for solving the multivariate algebraic equation. Our proposed procedure produces all exact solutions to the algebraic equation; therefore, these solutions are independent of the initial value and estimation algorithm. We conduct Monte Carlo simulations to investigate the characteristics of the maximum likelihood solutions.

翻译：在探索性因子分析中，模型参数通常通过最大似然方法进行估计。最大似然估计需要通过求解复杂的多元代数方程获得。由于该方程的解通常难以直接处理，一般使用连续优化方法（如牛顿-拉夫森方法）进行计算。然而，由于对数似然函数高度非凹，该过程得到的解不可避免地依赖于估计算法和初始值。特别地，独特方差的估计可能出现零值或负值（即不适当解），此时最大似然估计可能严重不稳定。为深入探究最大似然估计的不稳定性问题，我们利用代数计算求解多元代数方程的精确解。本文提出了一种基于代数计算的计算高效算法，该算法专门针对最大似然因子分析进行了优化。具体而言，我们采用了求解多元代数方程的强大工具——格罗布纳基和柱形代数分解。我们提出的方法能生成代数方程的所有精确解，因此这些解与初始值和估计算法无关。我们通过蒙特卡洛模拟研究了最大似然解的特性。