A theorem on non-associative algebras and its application to differential equations

Abstract

For a given field F, the set of F-algebras (resp. commutative F-algebras) of arity n≥2 and F-dimension m can be identified with the mⁿ⁺¹ (resp. m(^m+n−1 _n)) dimensional F-affine space S of structure coefficients. We show: If F is algebraically closed, then there exists an affine subvariety A of S with A≠S, which is defined over the prime field of F, such that all F-algebras corresponding to the points of S-A posses precisely n^m−1 idempotent elements ≠0 and fail to have nil potent elements ≠0. This implies for a system of ordinary differential equations

$$\left( * \right)\dot X_i = D_i \left( {X_l ,..,X_m } \right),i = l,..,m,$$

with D_i(X_i,...,X_m)∈ℂ[X₁,...,X_m] homogeneous polynomials of degree n: If the coefficients of the polynomials D_i, i=1,...,m, are algebraically independent over the field of rationals, then (*) possesses precisely n^m−1 ray solutions and fails to have a critical point other than the origin.