Tentative Schedule

## Computer Algebra Seminar No.26 (計算代数セミナー, 第26回) at Kobe Univ. (神戸大学)

Oct 2 (Sat), 2004. 10月2日 (土曜日), 11:00 --- 16:00

All talks will be presented in English.

Rouchdi Bahloul (Kobe University)
Local Grobner Fan --- Algorithmic and Polyhedral Approach , 11:00--12:30
Grobner fan (the equivalence relation by the initial ideals) is a fundamental object in computational algebraic geometry and in computational theory of D-modules. This object was firstly defined for a given homogeneous ideal in a (skew) polynomial ring. Is there a Grobner fan for ideals in the ring of power series and D_0? If there exists a fan under a relevant definition, we call it local Grobner fan. Local Grobner fan was firstly introduced by Assi, Castro, Granger (2001), but they did not prove that "it is a polyhedral fan". We prove that the local Grobner fan is a polyhedral fan, which also yields some applications including an (implementable) algorihtm. This is a joint work with Nobuki Takayama.

Guenael Renault (Paris 6)
Computation of the Decomposition Group of a Triangular Ideal , 14:30--16:00
Let $I$ be a triangular ideal of $K[X_1 , \ldots , X_n]$ a multivariate polynomial ring. The symmetric group $S_n$ acts naturally over $K[X_1 , \ldots , X_n]$ by permutating the indeterminates. We are interested in computation of $Dec(I)$ the set of permutations which leaves $I$ globally invariant. Actually, this set is a group called the \textit{decomposition group} of $I$ consistently with the classical definition on prime ideals (see [D\'efinition 2 page 36] Bourbaki, Alg. Comm. Chapitres 5 \`a 7).
In the specific case where $I$ is a relations ideal of a separable irreducible polynomial $f$ of degree $n$, Anai, Noro and Yokoyama give in [MEGA94] an algorithm for $Dec(I)$ computation which is a symmetric representation of the Galois group of $f$. They bound by $O(n^4)$ the number of normal forms computations needed by their algorithm.
In this talk we will present an algorithm which includes the backtracking technique (see Butler, Fundamental algorithms for permutation groups, 1991) in order to compute a strong generating set of $Dec(I)$.
Then we will present a generalization of Theorem $5$ of [Anai, Noro and Yokoyama MEGA94] to the case of Galois ideal and we prove that the number of normal forms computations needed by our algorithm is bounded by $O(n^3)$ in the case where the ideal $I$ is pure Galois (A relations ideal is pure Galois).
This talk is based upon a paper of Abdeljaouad-Tej, Orange, Renault and Valibouze (To appear in AAECC journal).

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The computer algebra seminar is co-organized by
M.Noro, noro@math.kobe-u.ac.jp
N.Takayama, takayama@math.kobe-u.ac.jp
K.Yokoyama, yokoyama@math.kyushu-u.ac.jp