Geodesic
The conditions of a local minimum of the functions in several variablesand implicit functions
Trudy Instituta matematiki, Tome 18 (2010) no. 2, pp. 11-21

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It is considered the relation between the local minimum conditions for a real analytic function in several variables and the problem about the implicit functions defined by system of the analytical equations. 
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P. P. Zabreiko; A. V. Krivko-Krasko. The conditions of a local minimum of the functions in several variablesand implicit functions. Trudy Instituta matematiki, Tome 18 (2010) no. 2, pp. 11-21. http://geodesic.mathdoc.fr/item/TIMB_2010_18_2_a1/