Tuesday, July 23, 2013 - from 16:00 to 16:25
We will discuss recent developments concerned with the so-called fundamental theorem of algebra for quaternions established in the seminal paper of S.Eilenberg and I.Niven . In , we explicated this classical result by proving that the zero-set of unilateral quaternion polynomial consists of $p$ points and $s$ two-dimensional spheres (counted with multiplicities) where $p+2s$ is equal to the algebraic degree of polynomial considered. We will complement the latter result by describing an effective way of counting multiplicities of the components of zero-set relying on the concept of quasi-norm of canonical quaternion polynomial introduced in  and further investigated in ). To this end we construct a special factorization of the quasi-norm and relate it to the structure of the zero-set. Moreover, we will present an effective criterion of stability of unilateral quaternion polynomial based on the results of , , which yields a solution to a problem explicitly formulated in .