The Cayley-Menger determinant is irreducible for $n\geq 3$

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

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…

关于特征多项式系数的梯度

On the gradient of the coefficient of the characteristic polynomial

本文证明双变量凯莱-哈密顿定理，统一导出经典凯莱-哈密顿定理、吉拉德-牛顿恒等式及行列式与特征多项式系数的多项式规模ABP，并基于系数梯度构造出规模为原纪录三分之一、宽度一半的新ABP，首次在28年内改进该问题，同时回答了clow序列必要性的26年未决问题。