Irreducibility of the Cayley-Menger determinant, and of a class of related polynomials

TL;DR

This paper proves the irreducibility of a class of polynomials related to the Cayley-Menger determinant, generalizing previous results and confirming a key step in understanding geometric relations in simplices.

Contribution

It establishes the irreducibility of the polynomial  in full generality, extending prior partial results and providing new proofs for related determinants.

Findings

Proves irreducibility of  without restrictions on parameters.

Provides new proofs for irreducibility of the Cayley-Menger determinant.

Generalizes previous results by D'Andrea and Sombra.

Abstract

If $S$ is a given regular $n$-simplex, $n \ge 2$, of edge length $a$, then the distances $a_1$, $\cdots$, $a_{n+1}$ of an arbitrary point in its affine hull to its vertices are related by the fairly known elegant relation $\phi_{n+1} (a,a_1,\cdots,a_{n+1})=0$, where $$\phi = \phi_t (x, x_1,\cdots,x_{n+1}) = \left( x^2+x_1^2+\cdots+x_{n+1}^2\right)^2 - t\left( x^4+x_1^4+\cdots+x_{n+1}^4\right).$$ 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 $\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 in Sib. J. Math. 46, 71--76.