On the properties of finite-dimensional systems of nonlinear equations with multiple solutions

Systems of nonlinear equations of the form $F(x)=0$, whose Jacobean is degenerate on the solution, are considered. Definitions of the value of solution's multiplicity are introduced for such problems. A relationship between the values of solution's multiplicity and the values of the index of matrix pencils has been determined; comparing with known definitions of multiplicity has been conducted. Possible applications of the concepts introduced in the theory of differential-algebraic equations, as well as in the process of analysis of non-Morse stationary point of the function and the values of the system's solution multiplicity, which satisfy the stationary point, are discussed.