We analyze the quantum description of a free scalar field on the circle in the presence of an explicitly time dependent potential, also interpretable as a time dependent mass. Classically, the field satisfies a linear wave equation of the formξ − ξ ′′ + f (t)ξ = 0. We prove that the representation of the canonical commutation relations corresponding to the particular case of a massless free field (f = 0) provides a unitary implementation of the dynamics for sufficiently general mass terms, f (t). Furthermore, this representation is uniquely specified, among the class of representations determined by S 1 -invariant complex structures, as the only one allowing a unitary dynamics. These conclusions can be extended in fact to fields on the two-sphere possessing axial symmetry. This generalizes a uniqueness result previously obtained in the context of the quantum field description of the Gowdy cosmologies, in the case of linear polarization and for any of the possible topologies of the spatial sections.

A method is developed to compute the zeros of a quaternion polynomial with all terms of the form ok Xk. This method is based essentially in Niven's algorithm (11, which consists of dividing the polynomial by a characteristic polynomial associated to a zero. The information about the trace and the norm of the zero is obtained by an original idea which requires the companion matrix associated to the polynomial. The companion matrix is represented by a matrix with complex entries. Three numerical examples using Mathematics 2.2 version are given.

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We discuss the generalization of results on quaternionic polynomials to the octonionic polynomials. In contrast to the quaternions the octonionic multiplication is non-associative. This fact although introducing some difficulties nevertheless leads to some new results. For instance, the monic and non-monic polynomials do not have, in general, the same set of zeros.Concerning the zeros, it is shown that in the monic and non-monic cases they are not the same, in general, but they belong to the same set of conjugacy classes.Despite these difficulties created by the non-associativity, we obtain equivalent results to the quaternionic case with respect to the number of zeros and the procedure to compute them. Mathematics Subject Classification (2000). 11R52, 20G20.

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.