Distances from the Vertices of a Regular Simplex

Hajja, Mowaffaq; Hayajneh, Mostafa; Nguyen, Bach; Shaqaqha, Shadi

doi:10.1007/s00025-017-0689-1

Cited by 4 publications

(8 citation statements)

References 18 publications

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“…Therefore F 0 is reducible if and only if 2S 2 2 − 2S 4 = 0, i.e., if and only if n = 3, in which case S 4 = x 4 1 = (x 2 1 ) 2 = S 2 2 . In this case F 0 is as given in (9). If t = 2, then F is a quadratic in v with coefficients in k[x 1 , · · · , x n−2 , u].…”

Section: Lemma 31 Let K Be a Field And Letmentioning

confidence: 99%

“…where t ∈ k and t = 0. If t = 2, then f factors as in (9). If t = 2, then f is reducible if and only if t = 3 and k contains a primitive third root ω of 1.…”

Section: Lemma 31 Let K Be a Field And Letmentioning

confidence: 99%

“…This raises the question whether the Cayley-Menger determinant M of a general n-simplex is irreducible. It is proved in Theorem 3.7 that the answer is affirmative except in the single case n = 2, in which case M is the Heron polynomial given in (9). This result has been obtained earlier in [5] when k is R or C. We should note, however, that the result in [5] does not imply the results in this paper, and does not give any information about the irreducibility of the Cayley-Menger determinant of any of the five special families mentioned earlier.…”

Section: Introductionmentioning

confidence: 95%

“…(1) see [4]. The natural question whether this is (essentially) the only relation was posed and answered in the affirmative in [9]. However, the proof rests heavily on a result whose proof was not included therein, but was postponed for another paper, namely this one.…”

Section: Introductionmentioning

confidence: 99%

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Irreducibility of the Cayley–Menger determinant and of a class of related polynomials

Hajja¹,

Hayajneh

Nguyen

et al. 2017

Beitr Algebra Geom

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Abstract. If S is a given regular n-simplex, n ≥ 2, of edge length a, then the distances a 1 , · · ·, a n+1 of an arbitrary point in its affine hull to its vertices are related by the fairly known elegant relation φ n+1 (a, a 1 , · · · , a n+1 ) = 0, whereThe natural question whether this is essentially the only relation is answered positively by M. Hajja, M. Hayajneh, B. Nguyen, and Sh. Shaqaqha in a recently submitted paper entitled Distances from the vertices of a regular simplex. In that paper, the authors made use of the irreducibility of the polynomial φ in the case when n ≥ 2, t = n + 1, x = a = 0, and k = R, but supplied no proof, promising to do so in another paper that is turning out to be this one. It is thus the main aim of this paper to establish that irreducibility. In fact, we treat the irreducibility of φ without restrictions on t, x, a, and k. As a by-product, we obtain new proofs of results pertaining to the irreducibility of the general Cayley-Menger determinant that are more general than those established by C. D'Andrea and M. Sombra in Sib. J. Math. 46, 71-76.

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Section: Lemma 31 Let K Be a Field And Letmentioning

confidence: 99%

“…where t ∈ k and t = 0. If t = 2, then f factors as in (9). If t = 2, then f is reducible if and only if t = 3 and k contains a primitive third root ω of 1.…”

Section: Lemma 31 Let K Be a Field And Letmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Irreducibility of the Cayley–Menger determinant and of a class of related polynomials

Hajja¹,

Hayajneh

Nguyen

et al. 2017

Beitr Algebra Geom

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“…The distances t 1 , · · · , t n+1 between the vertices of a regular n-simplex S of edge length t 0 and an arbitrary point P in its affine hull are related by the elegant relation (n + 1) (1) see [6] for a very short proof, and see [13] for a proof that (1) is essentially the only relation that exists among the quantities t 0 , · · · , t n+1 , for fixed t 0 . Figure 1 below illustrates the case when d = 2, i.e., when the regular simplex is an equilateral triangle.…”

Section: Introductionmentioning

confidence: 99%

Pre-kites: simplices having a regular facet

2017

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Abstract. The investigation of the relation among the distances of an arbitrary point in the Euclidean space R n to the vertices of a regular n-simplex in that space has led us to the study of simplices having a regular facet. Calling an n-simplex with a regular facet an n-pre-kite, we investigate, in the spirit of [14], [10], [9], and [15], and using tools from linear algebra, the degree of regularity implied by the coincidence of any two of the classical centers of such simplices. We also prove that if n ≥ 3, then the intersection of the family of n-pre-kites with any of the four known special families is the family of n-kites, thus extending the result in [18]. A basic tool is a closed form of a determinant that arises in the context of a certain Cayley-Menger determinant, and that generalizes several determinants that appear in [9], [15], and [16]. Thus the paper is a further testimony to the special role that linear algebra plays in higher dimensional geometry.Mathematics Subject Classification (2010). Primary 52B11; Secondary 52B12, 52B15, 51M20, 52B10.

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