5 problems

Let $\mathcal{H}$ be a Coxeter arrangement on a finite-dimensional inner product space $V$, let $a\in V$, and let $R$ satisfy $\|a\|\leq R$. Assume that the number of hyperplanes o…

For $d\geq 6$, define … Here $\mathbb{R}_{\geq0}[a_1^2+a_2^2,a_1^2a_2^2]$ denotes the polynomials in the two displayed generators with nonnegative real coefficients. Nonnegative in…

Matching-coefficient negativity conjecture. The coefficient of $\mu$ in $Y$ is negative. This coefficient-level assertion is used to establish the predicted sign in type $D_n$; it…

Let $V$ be an $n$-dimensional space, let $\mathcal{H}$ be a Coxeter arrangement of type $A_n$ or $D_n$ in $V$, and let $a\in V$ satisfy $0\in\mathbb{B}(a,R)$. Assume that $a$ lies…

Let $\mathsc{A}$ be the Coxeter arrangement associated with a finite reflection group $W$. The Hessian rank conjecture. The Hessian rank condition holds for $\mathsc{A}$ if $W$ is…