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We study in detail the zero set of a regular function of a quaternionic oroctonionic variable. By means of a division lemma for convergent power series,we find the exact relation existing between the zeros of two octonionic regularfunctions and those of their product. In the case of octonionic polynomials, weget a strong form of the fundamental theorem of algebra. We prove that the sumof the multiplicities of zeros equals the degree of the polynomial and obtain afactorization in linear polynomials.

Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect