An algorithm for approximating implicit functions by polynomials without using higher order differentiability information

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 information, 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.