Construction of Octonionic Polynomials

Abstract

In a previous paper "[On Octonionic Polynomials", Advances in Applied Clifford Algebras, 17 (2), (2007), 245-258] we discussed generalizations of results on quaternionic polynomials to the octonionic polynomials. In this paper, we continue this generalization searching for methods to construct octonionic polynomials with a prescribed set of zeros.Two iterative methods, valid for the quaternions, are applied to construct octonionic polynomials with limited results. The non-associativity of the octonion product does not allow the prescribed set of zeros to be the set of zeros of the constructed polynomial. Nevertheless, we will show that one of the methods has some advantage relatively to the other.Finally, a closed form method is given to construct an octonionic polynomial with a prescribed set of zeros. This method requires the inversion of a block Vandermonde matrix. The necessary and sufficient conditions for the existence of the inverse are studied.

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

“…On the octonions, it was recently proved by Serôdio [28]. The existence of an infinite numbers of monic regular quaternionic polynomials with prescribed multiple isolated zeros was proved by Beck [1].…”

Bounds for the zeros of unilateral octonionic polynomials

“…Example 3.1. For this example we could have constructed an octonionic polynomial with a prescribed set of zeros (see [27] for how to construct an octonionic polynomial with prescribed conditions), but this would probably imply a messy set of coefficients. Since we are not interested in the zeros but in a bound for them, we prevailed the coefficients choosing them with integer imaginary parts.…”

“…More recently, Serôdio [26] proved that, despite belonging to the same set of conjugacy classes, the zeros of a non-monic octonionic polynomial do not coincide, in general, with the zeros of the corresponding monic polynomial. In [27] he searched for methods to construct octonionic polynomials with a prescribed set of zeros.…”