On Integrals of Ordinary Differential Equations Systems which are Representable in Finite Terms
Existing theories on resolvability of nonlinear differential equations systems in a finite terms are generalization of Galois theory and for this reason the list of elementary operations is subject of the contract. In the Stockholm lectures (1897) Painleve gave on the example of the equations of the 1st and 2nd order property which is common for all equations, solvable in elementary, special and abelian functions: the general solutions of these equations depend on integration constants algebraically. Thus, if we record algebraic properties of the common decision, we can allocate a class of all-usable transcendental functions. This statement can be inscribed in a circle of the theory of Galois, i.e. we can construct the theory for the differential equations without fixing of this list. We consider an arbitrary system of ordinary differential equations g1(x1,..., x'1)=0,..., here g1,... are polynomials from x1,x'1..., which coefficients lie in a field k of functions of a variable t, for example in k = C(t). This system has solutions in an algebraically closed field K, for example in the field of Puiseux series. We will assume that ideal p =(f1,...) of ring K[x1,... ] is simple and that there is a differentiation D of the ring the rational functions on affine variety V (p/K), which kernel is a field of integrals of the system. Coefficients of integrals generate a field over k. We will designate its transcendence degree as r and prove that there are r-parametrical group of automorphisms for the field of integrals. This theorem will be used for calculation of integrals of these equations.