Polynomials and degrees of maps in real normed algebras - Archive ouverte HAL
Article Dans Une Revue Communications in Mathematics Année : 2020

Polynomials and degrees of maps in real normed algebras

Takis Sakkalis
  • Fonction : Auteur correspondant

Résumé

Let A be the algebra of quaternions H or octonions O. In this manuscript an elementary proof is given, based on ideas of Cauchy and D’Alembert, of the fact that an ordinary polynomial f(t) ∈ A[t] has a root in A. As a consequence, the Jacobian determinant |J(f)| is always nonnegative in A. Moreover, using the idea of the topological degree we show that a regular polynomial g(t) over A has also a root in A. Finally, utilizing multiplication (∗) in A, we prove various results on the topological degree of products of maps. In particular, if S is the unit sphere in A and h1, h2 : S → S are smooth maps, it is shown that deg(h1 ∗ h2) = deg(h1) + deg(h2).
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hal-03664980 , version 1 (11-05-2022)

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Takis Sakkalis. Polynomials and degrees of maps in real normed algebras. Communications in Mathematics, 2020, Volume 28 (2020), Issue 1 (1), pp.43 - 54. ⟨10.2478/cm-2020-0004⟩. ⟨hal-03664980⟩
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