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In the article, the object of research is the development of a technology for improving the operational properties of cast parts from steel 45, obtained by the method of electroslag remelting. Based on the research results, the modes of heat treatment of the shaft-gear part in an aqueous medium water-oil, heat treatment to increase the service life of the product, and energy-saving melting technology by the method of electroslag remelting were developed. Particular attention is paid to the development and analytical implementation of the mathematical model of the developed technologies by the methods of linear algebra by drawing up the Lagrange interpolation polynomial. Functional dependencies have been obtained for determining the amount of oxygen in the melt depending on the diameter of the electrode and the aluminum content in this electrode for any diameter of the mold, the hardness of the shaft surface depending on its diameter, the oxygen content (amount) in the melt depending on the aluminum content in the electrode for different mold diameters. Based on the developed and analytically implemented mathematical models of the investigated technological process, it seems possible to simplify experimental studies. In other words, based on analytical studies, it is possible to predict the results of subsequent experiments. Based on the developed and analytically implemented mathematical models of the investigated technological process, it seems possible to simplify experimental studies. In other words, based on analytical studies, it is possible to predict the results of subsequent experiments.