Factor-Analysis of nonlinear mappings: p-regularity theory

The paper presents recent advances in p-regularity theory, which has been developing successfully for the last twenty years. The main result of this theory gives a detailed description of the structure of the zero set of an irregular nonlinear mapping. We illustrate the theory with an application to degenerate problems in different fields of mathematics, which substantiates the general applicability of the class of p-regular problems. Moreover, the connection between singular problems and nonlinear mappings is shown. Amongst the applications, the structure of p-factor-operators is used to construct numerical methods for solving degenerate nonlinear equations and optimization problems.