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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
Article

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−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.

Keywords

Differential Equation Ordinary Differential Equation Number Theory Algebraic Geometry Topological Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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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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