$P$-factor interpolation of solutions of an equation with a degenerate function

The paper considers a new method for interpolation of nonlinear functions on an interval, the so-called $p$-factor interpolation method. By using a Newton interpolation polynomial as an example, it is shown that, in the case of degeneration of the approximated function $f(x)$ in the solution, classical interpolation does not provide the necessary accuracy for finding an approximate solution to the equation $f(x)$ =0, in contrast to the nondegenerate regular case. In turn, the use of $p$-factor interpolation polynomials for approximating functions in order to obtain the desired approximate solution to the equation provides the necessary order of accuracy in the argument during the calculations. The results are based on constructions of $p$-regularity theory and the apparatus of $p$-factor operators, which are effectively used in the study of degenerate mappings.