Titles and Short Summaries of the Talks

14 Algebra
9:00–12:00
March 21st (Thu) Conference Room I
Final: 2013/2/7
21 Noriko Zaitsu (Eigakuin) The field highter dimension over R than the sedenions does not exist
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 10
Summary: The field in which the solution of x 2 = −1 is added to a real number field is a
complex number field. It is two dimensional expansion. Similarly if one origin which fills x 2 = −1
which is not ±i is attached to a complex number, in addition this is set to j, it means that it
was added automatically ij = k will also be two-dimensional expansion and will be the number of
Hamiltonians. Why its number of Hamiltonians is non-commutative field is shown by a diagram.
(figure.1k) Similarly, the number of Cayley can also be defined and it becomes a nonassociative
field. If the definition is given similarly, since the origin in which the direction of the sedenions does
not become settled, addition cannot be defined, it is not the field. (figure.2k)
22 Shinichi Tajima (Univ. of Tsukuba) ♯ Efficient symbolic computation of matrix polynomials with an extended
Katsuyoshi Ohara (Kanazawa Univ.)
Akira Terui (Univ. of Tsukuba)
Horner’s rule · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 10
Summary: The Horner’s Rule is well-known method for evaluating univariate polynomials ef-
ficiently. For calculating “matrix polynomials” by evaluating the polynomial with matrices, we
propose an efficient method by extending naive Horner’s rule. Our key idea is dividing computation
of the Horner’s rule into parts to reduce the number of matrix multiplications which dominates
the computing time of matrix polynomials with naive Horner’s rule. We have implemented our
new algorithm into computer algebra system Risa/Asir, and show that, with proper settings on
the degree of division of the Horner’s rule, not only the computing time but also the amount of
memory used for the computation have been decreased, compared to the calculation with naive
Horner’s rule. We also show an attempt for even more efficient computation of our new algorithm
with distributed/parallel computing facility of Risa/Asir.
23 Shinichi Tajima (Univ. of Tsukuba) ♯
Katsuyoshi Ohara (Kanazawa Univ.)
On structure of invariant subspaces for square matrix · · · · · · · · · · · · · · · 10
Summary: We discuss structure of invariant subspaces of square matrix. It is an application of
minimal annihilating polynomials of matrix.
24 Katsuyoshi Ohara (Kanazawa Univ.) ♯ A randomized algorithm for computing minimal annihilating polynomi-
Shinichi Tajima (Univ. of Tsukuba) als of square matrix · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 10
Summary: A fast method for computing minimal annihilating polynomials of square matrix is
developed. The method uses a randomized algorithm.
25 Shuzo Izumi (Kinki Univ.) ♯ A family of Artinian rings associated to a finite-dimensional vector
space of holomorphic functions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 10
Summary: Let Z ⊂ O(U) be a finite-dimensional vector space of holomorphic functions on an
open subset Ω ⊂ C n . Let fb ↓ denote the homogeneous term of the minimal degree of the Taylor
expansion of f ∈ Z centred at b ∈ Ω and put Zb ↓= Span C{fb ↓: f ∈ Z}. Then the natural
sesquilinear form on Zb ↓ ×Z is a non-degenerate one (de Boor–Ron). It is also known that Zb
is D-invariant i.e. closed with respect to partial differentiation on a non-empty analytically open
subset U ⊂ Ω. These imply that Zb has a natural structure of a ring such that π : Ob −→ Zb is a
retraction homomorphism. Thus we have associated a deformation of graded Artinian local rings
{Zb : b ∈ U} to Z. We make a remark on the types of such rings.