Search results for "Integer matrix"

showing 2 items of 2 documents

Splittings of Toric Ideals

2019

Let $I \subseteq R = \mathbb{K}[x_1,\ldots,x_n]$ be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal $I$ can be "split" into the sum of two smaller toric ideals. For a general toric ideal $I$, we give a sufficient condition for this splitting in terms of the integer matrix that defines $I$. When $I = I_G$ is the toric ideal of a finite simple graph $G$, we give additional splittings of $I_G$ related to subgraphs of $G$. When there exists a splitting $I = I_1+I_2$ of the toric ideal, we show that in some cases we can describe the (multi-)graded Betti numbers of $I$ in terms of the (multi-)graded Betti numbers of $I_1$ and $I_2$.

Binomial (polynomial)Betti numberPrime idealExistential quantificationCommutative Algebra (math.AC)01 natural sciencesCombinatoricsInteger matrixMathematics::Algebraic Geometry0103 physical sciencesFOS: MathematicsGraded Betti numbers; Graphs; Toric idealsMathematics - Combinatorics0101 mathematicsMathematics::Symplectic GeometryMathematicsAlgebra and Number TheorySimple graphIdeal (set theory)Mathematics::Commutative AlgebraGraded Betti numbers Graphs Toric ideals010102 general mathematicsMathematics::Rings and Algebras16. Peace & justiceMathematics - Commutative AlgebraSettore MAT/02 - AlgebraToric ideals13D02 13P10 14M25 05E40Settore MAT/03 - Geometria010307 mathematical physicsCombinatorics (math.CO)Graded Betti numbersGraphs
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Generalized centro-invertible matrices with applications

2014

Centro-invertible matrices are introduced by R.S. Wikramaratna in 2008. For an involutory matrix R, we define the generalized centro-invertible matrices with respect to R to be those matrices A such that RAR = A^−1. We apply these matrices to a problem in modular arithmetic. Specifically, algorithms for image blurring/deblurring are designed by means of generalized centro-invertible matrices. In addition, if R1 and R2 are n × n involutory matrices, then there is a simple bijection between the set of all centro-invertible matrices with respect to R1 and the set with respect to R2.

Centro-symmetric matrixSquare root of a 2 by 2 matrixApplied MathematicsInvolutory matrixINGENIERIA TELEMATICAMatrius (Matemàtica)Matrix ringMatrix multiplicationCombinatoricsMatrix (mathematics)Integer matrix2 × 2 real matricesCentro-invertible matrixMatrix analysisInvolutory matrixMATEMATICA APLICADAComputer Science::Distributed Parallel and Cluster ComputingMathematics
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