manuscripta mathematica

, Volume 21, Issue 2, pp 181–187 | Cite as

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

  • Helmut Röhrl


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


Differential Equation Ordinary Differential Equation Number Theory Algebraic Geometry Topological Group 
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Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • Helmut Röhrl
    • 1
  1. 1.Department of MathematicsUniversity of California at San DiegoLa JollaUSA

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