• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.711-725
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• 2008

We give an explicit construction of Grobner-Shirshov pairs and monomial bases for finite-dimensional irreducible representations of the simple Lie algebra $sp_4$. We also identify the monomial basis consisting of the reduced monomials with a set of semistandard tableaux of a given shape, on which we give a colored oriented graph structure.

• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.41-62
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• 2010

In this article, we discuss a Grobner basis algorithm related to the stability of algebraic varieties in the sense of Geometric Invariant Theory. We implement the algorithm with Macaulay 2 and use it to prove the stability of certain curves that play an important role in the log minimal model program for the moduli space of curves.

• Journal of the Korean Mathematical Society
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• v.37 no.1
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• pp.55-72
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• 2000

In this paper, we develop the Grobner-Shirshov basis theory for the representations of associative algebras by introducing the notion of Grobner-Shirshov pairs. Our result can be applied to solve the reduction problem in representation theory and to construct monomial bases of representations of associative algebras. As an illustration, we give an explicit construction of Grobner-Shirshov pairs and monomial bases for finite dimensional irreducible representations of the simple tie algebra sl$_3$. Each of these monomial bases is in 1-1 correspondence with the set of semistandard Young tableaux with a given shape.

• Bulletin of the Korean Mathematical Society
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• v.40 no.3
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• pp.425-435
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• 2003

In this paper, a new kind of matrices, i.e., $level-{\kappa}$ II-circulant matrices is considered. Algorithms for computing minimal polynomial of this kind of matrices are presented by means of the algorithm for the Grobner basis of the ideal in the polynomial ring. Two algorithms for finding the inverses of such matrices are also presented based on the Buchberger's algorithm.

• Bulletin of the Korean Mathematical Society
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• v.34 no.4
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• pp.595-601
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• 1997

Let $C_d$ be the rational curve of degree d in $P_k ^3$ given parametrically by $x_0 = u^d, X_1 = u^{d - 1}t, X_2 = ut^{d - 1}, X_3 = t^d (d \geq 4)$. Then the defining ideal of $C_d$ can be minimally generated by d polynomials $F_1, F_2, \ldots, F_d$ such that $degF_1 = 2, degF_2 = \cdots = degF_d = d - 1$ and $C_d$ is a set-theoretically complete intersection on $F_2 = X_1^{d-1} - X_2X_0^{d-2}$ for every field k of characteristic p > 0. For the proofs we will use the notion of Grobner basis.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.251-263
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• 2004

In this paper, algorithms for computing the minimal polynomial and the common minimal polynomial of resultant matrices over any field are presented by means of the approach for the Grobner basis of the ideal in the polynomial ring, respectively, and two algorithms for finding the inverses of such matrices are also presented. Finally, an algorithm for the inverse of partitioned matrix with resultant blocks over any field is given, which can be realized by CoCoA 4.0, an algebraic system over the field of rational numbers or the field of residue classes of modulo prime number. We get examples showing the effectiveness of the algorithms.

• Korean Journal of Mathematics
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• v.30 no.2
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• pp.187-198
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• 2022

In this paper, we introduce a new concept in number theory called ζ-factors associated with a positive integer n. Applications of ζ-factors are in the arrangement of the defining polynomials in cyclic n-roots algebraic system and are thoroughly investigated. More precisely, ζ-factors arise in the proofs of vanishing theorems in regard to associated prime factors of the system. Exact computations through concrete examples of positive dimensions for n = 16, 18 support the results.