NON-CLASSICAL AND HIDDEN SYMMETRIES OF NONLINEAR ALGEBRAIC EQUATIONS AND SYSTEMS

Abstract

Classical and non-classical symmetries of algebraic equations and systems of algebraic equations are considered. Transformations that preserve the form of some algebraic equations, as well as transformations that reduce the order of these equations, are described. It is shown that individual algebraic equations with hidden symmetries can be reduced to classical symmetric systems of algebraic equations by introducing a new additional variable. It has been established that symmetric systems of algebraic equations of mixed type, consisting of symmetric and antisymmetric polynomials, can be converted to simpler systems. A method is presented for solving non-classical symmetric systems of two algebraic equations that change places when the unknowns are rearranged. Algebraic equations containing the second iteration of a given polynomial are studied, which are reduced to non-classical symmetric systems of equations. Examples are given of solving specific algebraic equations and systems of such equations that admit explicit and hidden symmetries. In particular, nontrivial algebraic equations of the sixth and ninth degrees are considered, containing free parameters that admit solutions in radicals. Irrational equations are described, which, by introducing two new variables, are reduced to symmetric systems of algebraic equations.