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### Hermite normal forms and δ-vector [article]

Takayuki Hibi, Akihiro Higashitani, Nan Li
2011 arXiv   pre-print
., δ_d), where ∑_i=0^d δ_i ≤ 4, by means of Hermite normal forms of square matrices.  ...  First, we consider the Hermite normal forms (10) .  ...  In the second subsection, we will consider a special "one row" Hermite normal form, which allows us to have better results. 3.1. "One row" Hermite normal forms.  ...

### Bricklaying and the Hermite Normal Form

William J. Gilbert
1993 The American mathematical monthly
This basis will form the columns of one of the modified Hermite normal forms obtained from A.  ...  What is connection between the brickwork in Figure 3 and the Hermite normal form?  ...

### Bricklaying and the Hermite Normal Form

William J. Gilbert
1993 The American mathematical monthly
This basis will form the columns of one of the modified Hermite normal forms obtained from A.  ...  What is connection between the brickwork in Figure 3 and the Hermite normal form?  ...

### Hermite normal forms and δ-vectors

Takayuki Hibi, Akihiro Higashitani, Nan Li
2012 Journal of combinatorial theory. Series A
Hermite normal form Let δ(P) = (δ 0 , δ 1 , . . . , δ d ) be the δ-vector of an integral polytope P ⊂ R N of dimension d.  ...  We obtain our results by considering-by means of Hermite normal forms of square matrices-the classification of integral simplices with a given δ-vector (δ 0 , δ 1 , . . . , δ d ), where d i=0 δ i 4.  ...  The other hermit normal form takes the form with d 1 = 2, d 1 = 3, e 1 = 1, e 1 = 2, d 1 = 3 and * = 1.) First,we consider the Hermite normal forms A 4 .  ...

### Natural Density Distribution of Hermite Normal Forms of Integer Matrices [article]

Gerard Maze
2011 arXiv   pre-print
The Hermite Normal Form (HNF) is a canonical representation of matrices over any principal ideal domain.  ...  Distribution of Hermite normal forms We start this section by noticing that the pivots h iji of the HNF of a matrix A are determined by the greatest common divisor of the i × i minors of the matrix that  ...  normal forms are not uniformly distributed among the upper triangular matrices.  ...

### Rational invariants of scalings from Hermite normal forms

Evelyne Hubert, George Labahn
2012 Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation - ISSAC '12
The primary tools used are Hermite normal forms and their unimodular multipliers.  ...  Scalings form a class of group actions that have both theoretical and practical importance. A scaling is accurately described by an integer matrix.  ...  Thus for each A there exists a unimodular matrix V such that A · V is in Hermite normal form. Similar statements also hold for the row Hermite normal form.  ...

### Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix [article]

George Labahn, Vincent Neiger, Wei Zhou
2017 arXiv   pre-print
Given a nonsingular n × n matrix of univariate polynomials over a field K, we give fast and deterministic algorithms to compute its determinant and its Hermite normal form.  ...  Our algorithms are based on a fast and deterministic triangularization method for computing the diagonal entries of the Hermite form of a nonsingular matrix.  ...  In the latter case, the best-known example is the Hermite normal form, first defined by Hermite in 1851 in the context of triangularizing integer matrices  .  ...

### Complexity of the Havas, Majewski, Matthews LLL Hermite Normal Form Algorithm

Wilberd Van Der Kallen
2000 Journal of symbolic computation
A A = bG is eventually in upside down Hermite normal form, see the abstract.  ...  As they do not affect b, we may remove from G all columns that do not contribute a pivot to the Hermite normal form.  ...

### Hermite and Smith normal form algorithms over Dedekind domains

Henri Cohen
1996 Mathematics of Computation
We show how the usual algorithms valid over Euclidean domains, such as the Hermite Normal Form, the modular Hermite Normal Form and the Smith Normal Form can be extended to Dedekind rings.  ...  a modular version), and the Smith Normal Form algorithm.  ...  The main theorem of this section is that the notion of Hermite Normal Form can be extended to Dedekind domains.  ...

### Efficient parallelizations of Hermite and Smith normal form algorithms

Gerold Jäger, Clemens Wagner
2009 Parallel Computing
To avoid these problems, we develop parallelizations of Hermite and Smith normal form algorithms.  ...  Hermite and Smith normal form are important forms of matrices used in linear algebra. These terms have many applications in group theory and number theory.  ...  The purpose of this paper is to show efficient parallelizations of Hermite and Smith normal form computations with empirical evidence.  ...

### Random Integer Lattice Generation via the Hermite Normal Form

Gengran Hu, Lin You, Liang Li, Liqin Hu, Hui Wang
2021 Entropy
Hu et al. and present an improved generation algorithm for it via the Hermite normal form.  ...  For large number N, this "random integer lattice" is uniformly chosen from the set of all n × n Hermite normal forms with the determinant equal to N.  ...

### A Polynomial-time Algorithm to Compute Generalized Hermite Normal Form of Matrices over Z[x] [article]

Rui-Juan Jing, Chun-Ming Yuan, Xiao-Shan Gao
2016 arXiv   pre-print
Second, fast algorithms to compute Hermite normal forms of matrices over Z are used.  ...  Third, the complexity of the algorithm are guaranteed by a nice estimation for the degree and height bounds of the polynomials in the generalized Hermite normal form.  ...  Jianwei Li for providing us information on the complexity of computing Hermite normal forms.  ...

### A fast, deterministic algorithm for computing a Hermite Normal Form of a polynomial matrix [article]

George Labahn, Wei Zhou
2016 arXiv   pre-print
The method relies of a fast algorithm for determining the diagonal entries of its Hermite normal form, having as cost O^∼(n^ωs) operations with s the average of the column degrees of F.  ...  Given a square, nonsingular matrix of univariate polynomials F∈K[x]^n × n over a field K, we give a fast, deterministic algorithm for finding the Hermite normal form of F with complexity O^∼(n^ωd) where  ...  Hermite Normal Form In Section 3, we have shown how to efficiently determine the diagonal entries of the Hermite normal form of a nonsingular input matrix F ∈ K [x] n×n .  ...

### About solving linear integer programs through Hermite Normal Form decomposition [chapter]

J. Maublanc, A. Quilliot
1996 System Modelling and Optimization
This paper discusses the ways Hermite Normal Form decompositions may be used in order to solve integer linear programs.  ...  HERMITE NORMAL FORM AND DIOPHANTINE SYSTEMS Recall: Nannal Hennite Fann.  ...  First, we notice that if a given square matrix defines a fractional vertex of some polyedron, then setting this matrix in Hermite Normal Form means performing some change of variables which preserves the  ...

### Natural density distribution of Hermite normal forms of integer matrices

Gérard Maze
2011 Journal of Number Theory
The Hermite Normal Form (HNF) is a canonical representation of matrices over any principal ideal domain.  ...  Conclusion Numerical experiments indicate that for randomly looking integer matrices, their Hermite normal forms are not uniformly distributed among the upper triangular matrices.  ...  On the more specialized study of density of canonical form of matrices, let us mention the work of Evans  where the density of Smith normal form over the ring of integers of a local field is studied  ...
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