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Hermite normal forms and δ-vector
[article]

2011
*
arXiv
*
pre-print

., δ_d), where ∑_i=0^d δ_i ≤ 4, by means of

arXiv:1009.6023v2
fatcat:5g2pa2bba5hthi7v6kkcu6od3q
*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*. ...##
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Bricklaying and the Hermite Normal Form

1993
*
The American mathematical monthly
*

This basis will

doi:10.2307/2324456
fatcat:mthra2wew5cxlp24g6g7parhqu
*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*? ...##
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Bricklaying and the Hermite Normal Form

1993
*
The American mathematical monthly
*

This basis will

doi:10.1080/00029890.1993.11990395
fatcat:jtyaddwbqjbjdetzvgm3nw2u4m
*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*? ...##
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Hermite normal forms and δ-vectors

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 . ...

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

2011
*
arXiv
*
pre-print

The

arXiv:1009.4826v2
fatcat:w5muzesf6zb7bdqacijb3vqqhm
*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. ...##
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Rational invariants of scalings from Hermite normal forms

2012
*
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation - ISSAC '12
*

The primary tools used are

doi:10.1145/2442829.2442862
dblp:conf/issac/HubertL12
fatcat:ysxyagli4fcsnhpobc2u4nmyna
*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*. ...##
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Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix
[article]

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

arXiv:1607.04176v2
fatcat:jwkqtecvnneg7dfx3pwo2l744e
*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 [18] . ...##
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Complexity of the Havas, Majewski, Matthews LLL Hermite Normal Form Algorithm

2000
*
Journal of symbolic computation
*

A A = bG is eventually in upside down

doi:10.1006/jsco.2000.0374
fatcat:svc7qveby5gqznggnpjiq4z5ua
*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*. ...##
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Hermite and Smith normal form algorithms over Dedekind domains

1996
*
Mathematics of Computation
*

We show how the usual algorithms valid over Euclidean domains, such as the

doi:10.1090/s0025-5718-96-00766-1
fatcat:ntkddvvffzfs3comev4k7zqk4a
*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. ...##
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Efficient parallelizations of Hermite and Smith normal form algorithms

2009
*
Parallel Computing
*

To avoid these problems, we develop parallelizations of

doi:10.1016/j.parco.2009.01.003
fatcat:w5cs4mqnyfgpnoqqnvjy4wfexm
*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. ...##
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Random Integer Lattice Generation via the Hermite Normal Form

2021
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Entropy
*

Hu et al. and present an improved generation algorithm for it via the

doi:10.3390/e23111509
pmid:34828207
pmcid:PMC8620907
fatcat:2vnftjqa4jekzml4um7ruzhmny
*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. ...##
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A Polynomial-time Algorithm to Compute Generalized Hermite Normal Form of Matrices over Z[x]
[article]

2016
*
arXiv
*
pre-print

Second, fast algorithms to compute

arXiv:1601.01067v2
fatcat:su7bp4yppfbltmgoyrnrlaatfe
*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*. ...##
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A fast, deterministic algorithm for computing a Hermite Normal Form of a polynomial matrix
[article]

2016
*
arXiv
*
pre-print

The method relies of a fast algorithm for determining the diagonal entries of its

arXiv:1602.02049v1
fatcat:cze4zuiadnabfhrmibmy34tssa
*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 . ...##
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About solving linear integer programs through Hermite Normal Form decomposition
[chapter]

1996
*
System Modelling and Optimization
*

This paper discusses the ways

doi:10.1007/978-0-387-34897-1_59
fatcat:a64plxhiifadpj2tiku4i2lefq
*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 ...##
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Natural density distribution of Hermite normal forms of integer matrices

2011
*
Journal of Number Theory
*

The

doi:10.1016/j.jnt.2011.06.010
fatcat:n5yjyjpcy5dlpbyjzpnydwh4q4
*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 [6] where the density of Smith*normal**form*over the ring of integers of a local field is studied ...
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