24 problems

Let $p_{n+1}(z)=\sum_{i=1}^{k}Q_i(z)p_{n-i}(z)$ be a finite recurrence with fixed polynomial coefficients, and suppose $\deg p_n(z)=n$ for $n\in\mathbb N$. Let $\Xi_Q$ be the equim…

Let $\{p_n(z)\}_{n\in\mathbb N}$ be defined by … where $p_{-2}(z)=p_{-1}(z)=0$, $p_0(z)=1$, and $C\in\mathbb R$. Egecioglu–Redmond–Ryavec 3-conjecture. All polynomials in the seque…

Conjecture for sums of powers of quadratic polynomials. If $f\in\mathcal{QP}_n$, then $f$ has at most $2n-1$ real critical points. Moreover, if $\alpha$ is a negative integer, each…

Let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the unit disk and at least one root at $\beta$. For $P\in S(\beta)$, let…

For $n\geq 0$, let $s_{3,n}(x)$ denote the third-order Stirling subset polynomial. Third-order Stirling subset zero conjecture. The zeros of $s_{3,n}(x)$ lie in the left half-plane…

Let $d_2(B)$ be the function associated with the Heun equation, the confluent Heun equation, and the reduced confluent Heun equation, and let $D_k$ and $D_k^{[j],(k+j+1)}$ denote t…

Generalized Borcea variance conjecture. For every $p\geq 1$ and every such choice of weights,

Let $F$ be a complex polynomial of degree $n\geq 2$ with zeros $z_1,\dots,z_n$ and critical points $w_1,\dots,w_{n-1}$. Schoenberg's conjecture. … with equality if and only if all…

Let $F$ be a complex polynomial of degree $n\geq 2$ with zeros $z_1,\dots,z_n$ satisfying $|z_k|\leq 1$, and critical points $w_1,\dots,w_{n-1}$. Let $\gamma_n$ be the radius of th…

Schmeisser's conjecture. For every such choice of weights,

Critical-threshold conjecture. There exists a threshold $t_{\textrm{crit}}=t_{\textrm{crit}}(N)>0$ such that $x\mapsto q_N(x;t)$ has $N$ real simple zeros for $t>t_{\textrm{crit}}$…

Common-root conjecture. If $2\mid k$, then the only common roots of $\mathfrak{R}_{2k+1,p}(z)$ and $(pz)^{2k}-1$ are $z=\pm i/p$; if $2\nmid k$, then the two polynomials have no co…

Ramanujan-type polynomial zero conjecture. The polynomial $\mathfrak{R}_{2k+1,p}(z)$ has only the real zero $z=0$ of multiplicity $2$; all remaining zeros are non-real, lie on the…

Let $p(z)$ be a polynomial of degree $n$, let $\rho_{\min}:=\min_{\rho:p(\rho)=0}|\rho|$, and let $L_m$ be the lower bound in the inequality $L_m/\rho_{\min}\geq 1-\epsilon$. n-ind…

Let $\mathcal N_n$ and $\mathcal L_n$ denote the relevant families of Newman and Littlewood polynomials of degree $n$, and let $N$ and $\widetilde N$ count zeros in the open unit d…

Let $n\geq 1$, and let $(k,n)$ be a valid pair for Littlewood polynomials. A pair is Littlewood-admissible if some Littlewood polynomial of degree $n$ has exactly $k$ zeros in the…

Oura's conjecture. The analogous properties hold for the transforms of the normalized Eisenstein polynomials, namely:

At-most-two non-real zeros conjecture. Every polynomial $W_n(z)$ has at most two non-real zeros.

Let $T_n(x)$ be the polynomial defined by the exponential generating function … For polynomials with only real coefficients and only imaginary zeros, say that $f(x)$ separates…

For $N>0$, let … For $|q|<1$, the polynomial Sokal conjecture. The polynomial $P_N(z;q)$ can have only simple roots, separated in modulus by at least the factor $|q|$. This is the…

Alternating-location conjecture. If $d$ is even, all the zeros of $G(d;z)$ in $A_1$ are the zeros on the unit circle $\mathcal C_0$ given in that proposition. If $d$ is odd, the ze…

Zero-separation conjecture. For every positive integer $d$, the zeros of $G(d;z)$ form three groups separated by the sets $A_1$, $A_2$, and $A_3$, except that when $6$ divides $d$…

Let $\mathfrak d(z)$ be an exactly solvable operator, and let $\{S_n(z)\}_{n\in\mathbb N,\,n\geq n_0}$ be its eigenpolynomials with $\deg S_n(z)=n$. Let $\mathfrak S_\mu$ denote th…

Let $\mathfrak d(z)$ be a degenerate Lamé operator, meaning that its leading coefficient satisfies $\deg Q_k<k+r$, and let $N_0$ be any positive integer. Unbounded-root conjecture.…