Reducing the branching equation to a polynomial 

Aleksandr N. Grin, Candidate of physical and mathematical sciences, associate professor of the subdepartment of mathematics, Mozhaisky Military Space Academy (13 Zhdanovskaya street, St. Petersburg, Russia) E-mail: al-grinl@yandex.ru
Marina S. Rodionova, Candidate of pedagogical sciences, associate professor of the sub-department of mathematics, Mozhaisky Military Space Academy (13 Zhdanovskaya street, St. Petersburg, Russia) E-mail: margo-m@bk.ru
Ekaterina A. Shakhova, Candidate of engineering Sciences, associate professor of the sub-department of mathematics, Mozhaisky Military Space Academy (13 Zhdanovskaya street, St. Petersburg, Russia) E-mail: rinagrits2012@mail. ru

Backgound. The study of the local topological structure of the set of solutions of a nonlinear equation, as a rule, is reduced to solving a finite-dimensional branching equation constructed from the original equation. However, for the operator that defines the branching equation, it is often only possible to obtain a Taylor polynomial of any degree. What this degree should be for the local topological structure of the set of solutions of the truncated equation to be equivalent to the structure of solutions of the true equation is unknown. The purpose of this study is to solve this problem. Materials, methods, and results. The results of the work are based on the theory of singularities of differentiable maps developed by the works of R. Thoma and J. Meser. The Taylor polynomial of degree r, constructed by the branching equation, defines an r-stream of maps, that is, a class of maps having identical Taylor polynomials of degree r. A jet is called r-sufficient if any two representatives of this jet have the same local topological structure of the solution sets in the vicinity of the critical point. To establish the r-sufficiency of the jet, we construct a polynomial equation called the homological equation. The properties of the solutions of this equation allow us to determine whether the jet is r-sufficient or not. Since the operator defining the branching equation and its Taylor polynomial of degree r belong to the same r-jet, then, having established the r-sufficiency of the jet, we can state that the local structures of the set of solutions of the branching equation and the truncated equation are equivalent. Conclusion. The use of the homological equation makes it possible to reduce the study of the local structure of the set of solutions of the branching equation to the study of the set of solutions of the polynomial equation, which is determined by the Taylor polynomial.

germ of the mapping, r-stream of the mapping, local sufficiency of the rstream, infinitesimal stability of the germ of the mapping, homological equation

Grin A.N., Rodionova M.S., Shakhova E.A. Reducing the branching equation to a polynomial. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fizikomatematicheskie nauki = University proceedings. Volga region. Physical and mathematical sciences. 2024;(4):46–52. (In Russ.). doi: 10.21685/2072-3040-2024-4-4