## A remark about factorizing GCD-type (2008)

### BibTeX

@MISC{Luque08aremark,

author = {Jean-gabriel Luque},

title = {A remark about factorizing GCD-type},

year = {2008}

}

### OpenURL

### Abstract

We show that a very elementary trick allows to generalize a theorem of Linström to higher determinants. As a special case, one recovers some results due to Lehmer and Haukkannen on hyperdeterminants of GCD matrices. 1

### Citations

402 |
Introduction to Analytic Number Theory
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- 1976
(Show Context)
Citation Context ...long in a factor closed set (i.e. all the factors of an element belong to the set) as a product of Euler’s totient. The interest of this kind of equalities lies in its links with arithmetic functions =-=[1]-=- and in particular multiplicative functions (see [7, 8] for interesting remarks about the last notion). During the last century, many generalizations of the Smith theorem have been investigated. One o... |

49 | Advanced determinant calculus: a complement, Linear Algebra Appl
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(Show Context)
Citation Context ...ince the end of the nineteen century, it is known that some determinants, with entries depending only of the gcd of the indices, factorize. Readers interested by the story of the problem can refer to =-=[13]-=- and [2]. The subject is born in 1876, in an article of Smith [15] which has evaluated the determinant of a GCD matrix whose entries belong in a factor closed set (i.e. all the factors of an element b... |

13 |
On the value of a certain arithmetical determinant
- SMITH
(Show Context)
Citation Context ...inants, with entries depending only of the gcd of the indices, factorize. Readers interested by the story of the problem can refer to [13] and [2]. The subject is born in 1876, in an article of Smith =-=[15]-=- which has evaluated the determinant of a GCD matrix whose entries belong in a factor closed set (i.e. all the factors of an element belong to the set) as a product of Euler’s totient. The interest of... |

10 |
Determinants on semilattices
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(Show Context)
Citation Context ...he set) as a product of certain functions evaluated in terms of Euler’s totient. The fact that these determinants factorize can be seen as a particular case of a very elegant theorem due to Lindström =-=[11]-=- which evaluates the determinant of GCDmatrices whose indices are taken in a meet semilattice (i.e. a poset such that each pair admits a greatest lower bound). An other way to generalize Smith identit... |

8 |
On the theory of permutants
- Cayley
(Show Context)
Citation Context ...ional analogue of the Lindström theorem in Section 3. 2 Hyperdeterminants and F-determinants The question of extending the notion of determinant to higher dimensional arrays has been raised by Cayley =-=[5, 6]-=- few after he introduced the modern notation as square arrays [4]. The simplest generalization is defined for a kth order tensor on an n-dimensional space M = (Mi1,···,ik )1≤i1,···,ik≤n by the alterna... |

8 |
Spatial matrices and their application
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- 1960
(Show Context)
Citation Context ...lize Smith identity consists on computing multidimensional analogous. Lehmer [9] has given in 1930 the first multi-indexed version of the Smith determinant. Other related computation are collected in =-=[16, 17]-=-. More recently, Haukkannen [12] gave a hyperdeterminantal generalization of the equality due to Beslin and Ligh (note that in the same paper he has computed a multidimensional version of the equality... |

6 |
The Determinants of GCD Matrices
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(Show Context)
Citation Context ...y generalizations of the Smith theorem have been investigated. One of the way to extend this result 1consists on changing the set of the indices of the matrices. The more general result is due to Li =-=[10]-=- in 1990, which gives the value of GCD determinant for an arbitrary set of indices. The present paper is not devoted to this more general determinants which can not be easily factorized. Beslin and Li... |

6 |
Introduction to the theory of multidimensional matrices (in Russian), Nukova hyperdeterminants 32
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(Show Context)
Citation Context ...lize Smith identity consists on computing multidimensional analogous. Lehmer [9] has given in 1930 the first multi-indexed version of the Smith determinant. Other related computation are collected in =-=[16, 17]-=-. More recently, Haukkannen [12] gave a hyperdeterminantal generalization of the equality due to Beslin and Ligh (note that in the same paper he has computed a multidimensional version of the equality... |

5 |
Another Generalization of Smith's Determinant
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(Show Context)
Citation Context ... 1990, which gives the value of GCD determinant for an arbitrary set of indices. The present paper is not devoted to this more general determinants which can not be easily factorized. Beslin and Ligh =-=[3]-=- gave a factorization of such a determinant when the indices run in a gcdclosed set (i.e. the gcd of two elements belongs in the set) as a product of certain functions evaluated in terms of Euler’s to... |

3 |
On the theory of determinants, Trans
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(Show Context)
Citation Context ...nants and F-determinants The question of extending the notion of determinant to higher dimensional arrays has been raised by Cayley [5, 6] few after he introduced the modern notation as square arrays =-=[4]-=-. The simplest generalization is defined for a kth order tensor on an n-dimensional space M = (Mi1,···,ik )1≤i1,···,ik≤n by the alternated sum DetM = 1 n! ∑ σ=(σ1,···,σk)∈S k n sign(σ)M σ , where sign... |

3 |
P-Way Determinants, with an Application to Transvectants
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(Show Context)
Citation Context ...M σ . (1) σ=(Id,σ2,···,σk)∈S k n When k is even the two notions coincide but for k odd, only Det1 does not vanish. This is a special cases of the ”less-than-full-sign” determinants theory due to Rice =-=[14]-=-. If F denotes a map from Sk−2 n to a commutative ring. One defines a more general object, the F-determinant DetF(M) of M, by DetF(M) = ∑ sign(σ2)F(σ3, . . ., σk) ∏ Miσ2(i)...σk(i). σ=(σ2,···,σk)∈S k−... |

3 |
Higher-dimensional GCD matrices
- Haukkanen
- 1992
(Show Context)
Citation Context ...computing multidimensional analogous. In 1930 Lehmer gave [9] the first multi-indexed version of the Smith’s determinant. Other related computation are collected in [16, 17]. More recently, Haukkanen =-=[12]-=- gave a hyperdeterminantal generalization of the equality due to Beslin and Ligh (note that in the same paper he computed a multidimensional version of the equality of Li [10]). We will see in Section... |

2 | GCD matrices, posets and non-intersecting paths, Linear Multilinear Algebra (to appear). math.CO/0406155. (p
- Altini¸sik, Sagan, et al.
(Show Context)
Citation Context ...end of the nineteen century, it is known that some determinants, with entries depending only of the gcd of the indices, factorize. Readers interested by the story of the problem can refer to [13] and =-=[2]-=-. The subject is born in 1876, in an article of Smith [15] which has evaluated the determinant of a GCD matrix whose entries belong in a factor closed set (i.e. all the factors of an element belong to... |

1 |
Addition of 1, Séminaire Lotharingien, Mars 04. 8p. 6 A. Lascoux, Multiplicative functions, http://www.combinatorics.net/lascoux/courses/dvi ps/Moebiusps.rar
- Lascoux
(Show Context)
Citation Context ... an element belong to the set) as a product of Euler’s totient. The interest of this kind of equalities lies in its links with arithmetic functions [1] and in particular multiplicative functions (see =-=[7, 8]-=- for interesting remarks about the last notion). During the last century, many generalizations of the Smith theorem have been investigated. One of the way to extend this result 1consists on changing ... |

1 |
The p dimensional analogue of Smith’s determinant
- Lehmer
- 1930
(Show Context)
Citation Context ...es are taken in a meet semilattice (i.e. a poset such that each pair admits a greatest lower bound). An other way to generalize Smith identity consists on computing multidimensional analogous. Lehmer =-=[9]-=- has given in 1930 the first multi-indexed version of the Smith determinant. Other related computation are collected in [16, 17]. More recently, Haukkannen [12] gave a hyperdeterminantal generalizatio... |

1 |
Addition of 1
- Lascoux
(Show Context)
Citation Context ... an element belong to the set) as a product of Euler’s totient. The interest of this kind of equalities lies in its links with arithmetic functions [1] and in particular multiplicative functions (see =-=[7, 8]-=- for interesting remarks about the last notion). During the last century, many generalizations of Smith’s theorem have been investigated. One of the way to extend 1this result consists on changing th... |

1 |
Multiplicative functions, http://www.combinatorics.net/lascoux/courses/dvi ps/Moebiusps.rar
- Lascoux
(Show Context)
Citation Context ... an element belong to the set) as a product of Euler’s totient. The interest of this kind of equalities lies in its links with arithmetic functions [1] and in particular multiplicative functions (see =-=[7, 8]-=- for interesting remarks about the last notion). During the last century, many generalizations of Smith’s theorem have been investigated. One of the way to extend 1this result consists on changing th... |