Geodesic
On the properties of finite-dimensional systems of nonlinear equations with multiple solutions
Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika, no. 11 (2009), pp. 20-36

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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. 
@article{VCHGU_2009_11_a2,
     author = {M. V. Bulatov and V. F. Chistyakov},
     title = {On the properties of finite-dimensional systems of nonlinear equations with multiple solutions},
     journal = {Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika},
     pages = {20--36},
     publisher = {mathdoc},
     number = {11},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VCHGU_2009_11_a2/}
}
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M. V. Bulatov; V. F. Chistyakov. On the properties of finite-dimensional systems of nonlinear equations with multiple solutions. Vestnik Chelyabinskogo Gosudarstvennogo Universiteta. Matematika, Mekhanika, Informatika, no. 11 (2009), pp. 20-36. http://geodesic.mathdoc.fr/item/VCHGU_2009_11_a2/