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Bricklaying and the Hermite Normal Form
"... We describe a geometric interpretation of the Hermite normal form of an integer matrix in terms of tilings by bricks. A nonsingular integer n × n matrix A generates an integer lattice, L, in nspace. The lattice points are the integer linear combinations of the columns of A. Figure 1 shows such a l ..."
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We describe a geometric interpretation of the Hermite normal form of an integer matrix in terms of tilings by bricks. A nonsingular integer n × n matrix A generates an integer lattice, L, in nspace. The lattice points are the integer linear combinations of the columns of A. Figure 1 shows such a
Rational invariants of scalings from Hermite normal forms.
, 2012
"... Scalings form a class of group actions on affine spaces that have both theoretical and practical importance. A scaling is accurately described by an integer matrix. Tools from linear algebra are exploited to compute a minimal generating set of rational invariants, trivial rewriting and rational sect ..."
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Cited by 6 (3 self)
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sections for such a group action. The primary tools used are Hermite normal forms and their unimodular multipliers. With the same line of ideas, a complete solution to the scaling symmetry reduction of a polynomial system is also presented.
Fast Computation of Hermite Normal Forms of Random Integer Matrices
"... This paper is about how to compute the Hermite normal form of a random integer matrix in practice. We propose significant improvements to the algorithm by Micciancio and Warinschi, and extend these techniques to the computation of the saturation of a matrix. Tables of timings confirm the efficiency ..."
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Cited by 7 (0 self)
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This paper is about how to compute the Hermite normal form of a random integer matrix in practice. We propose significant improvements to the algorithm by Micciancio and Warinschi, and extend these techniques to the computation of the saturation of a matrix. Tables of timings confirm the efficiency
Soving AX = B using the Hermite normal form
, 2011
"... matrices of size m × n, n × 1, m × 1, respectively, with A nonzero. One classical method, described in M. Newman’s book ([2, page 36]) uses the Smith Normal Form of A. ..."
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matrices of size m × n, n × 1, m × 1, respectively, with A nonzero. One classical method, described in M. Newman’s book ([2, page 36]) uses the Smith Normal Form of A.
Preconditioning of Rectangular Polynomial Matrices for Efficient Hermite Normal Form Computation
 In Proceedings of ISSAC'95
, 1995
"... We present a Las Vegas probabalistic algorithm for reducing the computation of Hermite normal forms of rectangular polynomial matrices. In particular, the problem of computing the Hermite normal form of a rectangular m \Theta n matrix (with m ? n) reduces to that of computing the Hermite normal form ..."
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Cited by 7 (5 self)
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We present a Las Vegas probabalistic algorithm for reducing the computation of Hermite normal forms of rectangular polynomial matrices. In particular, the problem of computing the Hermite normal form of a rectangular m \Theta n matrix (with m ? n) reduces to that of computing the Hermite normal
A LOWER TRIANGULAR HERMITE NORMAL FORM FOR PROJECTIONREGULAR LATTICE
, 2005
"... The structure of lattice rules has been studied using two different approaches. One of them is based on the generator matrix B of the dual of the integration lattice while the other approach is based on the representation of lattice rules in D − Z form. The former approach has previously made the as ..."
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the assumption that the Hermite normal form of the matrix B is upper triangular. However, for the special case of projectionregular rules in which the principal projections have the maximum possible number of distinct quadrature points, it is possible to specify a unique upper triangular matrix Z
A Linear Space Algorithm for Computing the Hermite Normal Form
 Proceedings ISSAC 2001, Lecture Notes in Computer Sci., 2146
, 2001
"... Computing the Hermite Normal Form of an n n integer matrix using the best current algorithms typically requires O(n 3 log M) space, where M is a bound on the entries of the input matrix. Although polynomial in the input size (which is O(n 2 log M)), this space blowup can easily become a seriou ..."
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Cited by 14 (2 self)
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Computing the Hermite Normal Form of an n n integer matrix using the best current algorithms typically requires O(n 3 log M) space, where M is a bound on the entries of the input matrix. Although polynomial in the input size (which is O(n 2 log M)), this space blowup can easily become a
on Complexity of the Havas, Majewski, Matthews LLL Hermite Normal Form Algorithm
"... We consider the complexity of the LLL HNF algorithm (Havas et al., 1998, Algorithm 4). This algorithm takes as input an m by n matrix G of integers and produces as output a matrix b ∈ GLm(Z) so that A = bG is in Hermite normal form (upside down). The analysis is similar to that of an extended LLL al ..."
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We consider the complexity of the LLL HNF algorithm (Havas et al., 1998, Algorithm 4). This algorithm takes as input an m by n matrix G of integers and produces as output a matrix b ∈ GLm(Z) so that A = bG is in Hermite normal form (upside down). The analysis is similar to that of an extended LLL
Extended gcd and Hermite normal form algorithms via lattice basis reduction
 Experimental Mathematics
, 1998
"... Extended gcd calculation has a long history and plays an important role in computational number theory and linear algebra. Recent results have shown that finding optimal multipliers in extended gcd calculations is difficult. We present an algorithm which uses lattice basis reduction to produce small ..."
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Cited by 33 (6 self)
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small integer multipliers x1,..., xm for the equation d = gcd (d1,..., dm) = x1d1 + · · · + xmdm, where d1,..., dm are given integers. The method generalises to produce small unimodular transformation matrices for computing the Hermite normal form of an integer matrix. 1
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