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

  • Mowaffaq Hajja
  • Mostafa Hayajneh
  • Bach Nguyen
  • Shadi Shaqaqha
Original Paper
  • 54 Downloads

Abstract

If S is a given regular n-simplex, \(n \ge 2\), of edge length a, then the distances \(a_1, \ldots , a_{n+1}\) of an arbitrary point in its affine hull to its vertices are related by the fairly known elegant relation \(\phi _{n+1} (a,a_1,\ldots ,a_{n+1})=0\), where
$$\begin{aligned} \phi = \phi _t (x, x_1,\ldots ,x_{n+1}) = \left( x^2+x_1^2+\cdots +x_{n+1}^2\right) ^2 - t\left( x^4+x_1^4+\cdots +x_{n+1}^4\right) . \end{aligned}$$
The natural question whether this is essentially the only relation was recently and positively answered by M. Hajja, M. Hayajneh, B. Nguyen, and Sh. Shaqaqha. The authors made use of the irreducibility of the polynomial \(\phi \) in the case when \(n \ge 2\), \(t=n+1\), \(x= a \ne 0\), and \(k = \mathbb {R}\), 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 \(\phi \) 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.

Keywords

Cayley–Menger determinant Circumscriptible simplex Discriminant Homogeneous polynomial Irreducible polynomial Isodynamic simplex Isogonic simplex Orthocentric simplex Pompeiu’s theorem Pre-kites Quadratic polynomial Regular simplex Symmetric polynomial Tetra-isogonic simplex Volume of a simplex 

Mathematics Subject Classification

12E05 12E10 112D05 

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Copyright information

© The Managing Editors 2017

Authors and Affiliations

  • Mowaffaq Hajja
    • 1
  • Mostafa Hayajneh
    • 2
  • Bach Nguyen
    • 3
  • Shadi Shaqaqha
    • 2
  1. 1.Philadelphia UniversityAmmanJordan
  2. 2.Yarmouk UniversityIrbidJordan
  3. 3.Louisiana State UniversityBaton RougeUSA

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