We consider an equation of multiple variables in which a partial derivative does not vanish at a point. The implicit function theorem provides a local existence and uniqueness of the function for the equation. In this paper, we propose an algorithm to approximate the function by a polynomial without using higher-order differentiability, which depends essentially on integrability. Moreover, we extend the method to a system of equations if the Jacobian determinant does not vanish. This is a robust method for implicit functions that are not differentiable to higher-order. Additionally, we present two numerical experiments to verify the theoretical results.

翻译：考虑一个多变量方程，其中某一偏导数在某点处非零。隐函数定理保证了该方程存在局部唯一解。本文提出了一种无需高阶可微性即可用多项式逼近该函数的算法，该算法本质上依赖于可积性。此外，我们将此方法推广至雅可比行列式非零的方程组情形。该方法对于非高阶可微的隐函数具有鲁棒性。最后，我们给出两个数值实验以验证理论结果。