Cayley-Menger行列式及一类相关多项式的不可约性
Irreducibility of the Cayley-Menger determinant, and of a class of related polynomials
摘要：
如果$S$是一个给定的正则$n$ -单形，$n \ge 2$，边长为$a$，则其仿射壳中任意一点到其顶点的距离$a_1$，$\cdots$，$a_{n+1}$由众所周知的优美关系$ϕ_{n+1} (a,a_1,\cdots,a_{n+1})=0$相关，其中$$ϕ= ϕ_t (x, x_1,\cdots,x_{n+1}) = \left( x^2+x_1^2+\cdots+x_{n+1}^2\right)^2 - t\left( x^4+x_1^4+\cdots+x_{n+1}^4\right).$$这是否是本质上唯一的关系的自然问题由M肯定地回答。Hajja，M.哈亚杰奈湾Nguyen和Sh. Shaqaqha在最近提交的一篇题为“距离顶点的正规单纯形。“在那篇论文中，作者在$n \ge 2$、$t=n+1$、$x= a \ne 0$和$k = \mathbb{R}$的情况下使用了多项式$ϕ$的不可约性，但没有提供证明，承诺在另一篇论文中这样做，结果就是这篇论文。因此，本文的主要目的是建立不可约性。事实上，我们处理$ϕ$的不可还原性，而不限制$t$、$x$、$a$和$k$。作为副产品，我们得到了关于一般Cayley-Menger行列式的不可约性的结果的新证明，这些结果比C. D 'Andrea和M.在Sib的阴影。《数学杂志》，46，71- 76。
英文摘要：
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 $ϕ_{n+1} (a,a_1,\cdots,a_{n+1})=0$, where $$ϕ= ϕ_t (x, x_1,\cdots,x_{n+1}) = \left( x^2+x_1^2+\cdots+x_{n+1}^2\right)^2 - t\left( x^4+x_1^4+\cdots+x_{n+1}^4\right).$$ 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 $ϕ$ 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 $ϕ$ 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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