Bach Nguyen - Academia.edu (original) (raw)

Papers by Bach Nguyen

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2017

If S is a given regular n-simplex, n ≥ 2, of edge length a, then the distances a 1 , • • •, a n+1... more 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, where φ = φ t (x, x 1 , • • • , x n+1) = x 2 + x 2 1 + • • • + x 2 n+1 2 − t x 4 + x 4 1 + • • • + x 4 n+1. The 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

Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2017

If S is a given regular n-simplex, n ≥ 2, of edge length a, then the distances a 1 , • • •, a n+1... more 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, where φ = φ t (x, x 1 , • • • , x n+1) = x 2 + x 2 1 + • • • + x 2 n+1 2 − t x 4 + x 4 1 + • • • + x 4 n+1. The 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