$p$th-order approximation of the solution set of nonlinear equations

Given a system of nonlinear equations, a formula is derived for the family of its approximate solutions of up to the pth order of smallness. A formula approximating an implicit function up to the third order of smallness is presented. Iterative methods converging with the $p$th order are constructed for solving systems of nonlinear equations. These results are extended to the degenerate case. Examples of applying the results are given.