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

If S is a given regular n-simplex, n≥ 2 , of edge length a, then the distances a1, … , an+1 of an arbitrary point in its affine hull to its vertices are related by the fairly known elegant relation ϕn+1(a, a1, … , an+1) = 0 , where ϕ=ϕt(x,x1,…,xn+1)=(x2+x12+⋯+xn+12)2-t(x4+x14+⋯+xn+14).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 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. © 2017, The Managing Editors.