Speaker:

Giorgi Khimshiashvili

Tuesday, July 23, 2013 - from 16:00
to 16:25

Abstract:

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 [2].
In [3], 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 [5] and further investigated
in [6]). 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
[3], [4], which yields a solution to a problem
explicitly formulated in [1].