8 problems

Resultant stability conjecture. For any positive integers $n,m$ such that $n\leq m$, and any $a>0$, the resultant $\operatorname{res}_z(P_n(s,z),P_m(s+a,z))$ is stable.

Let $\{f_k\}_{k=0}^{n}$ be a finite Pólya frequency sequence of order $d\geq r$, and let $P_n^r(z)$ be the higher-order Toeplitz determinant polynomial defined in the preceding con…

Hurwitz stability conjecture. The polynomial $T_{n,r}(x)$ is Hurwitz stable for all $r\ge 2$ and $n\in\mathbb{N}$. The sequence is introduced after Hurwitz stability has been estab…

Let $r(x)=\sqrt{\frac{1+x}{1-x}}$ and define $\mathscr{R}_n(x)=\sum_{k=0}^{2n-1}\mathscr{R}(n,k)x^k$ by … The recurrence is … with $\mathscr{R}_0(x)=1$ and $\mathscr{R}_1(x)=x$. Hu…

Algebraic-specification conjecture. For every admissible $\delta$, there is a quasi-admissible $\delta'$ with $\delta'>\delta$ that can be achieved by algebraically specified…

Let $\mathcal{P} = \{Y_k(x)\}_{k=0}^\infty$, where $Y_k$ is the $k$-th Bessel polynomial, with … The extended Turán expression $\mathscr{T}_k^{(n)}(\mathcal{P};x)$ is defined for…

Let $PF_r$ be the class of sequences whose infinite Toeplitz matrix has all minors of order at most $r$ non-negative. For $r\geq2$, define … Here $(x)_k$ is the rising factorial, a…

Let $\{f_k\}_{k=0}^{n}$ be a non-negative sequence, let $\alpha,\beta>0$, and define … A real polynomial is Hurwitz stable when all its zeros have negative real part. The Hurwitz-s…