21 problems

Let $S_n$ be the set of monic complex polynomials of degree $n\ge 2$ whose zeros lie in the closed unit disk. For $p\in S_n$, define … Call $p$ extremal for Sendov's conjecture if…

Let $\beta\in\bar{D}$ and let $S(n,\beta)$ be the set of monic complex polynomials in $S_n$ having at least one zero at $\beta$. For $\alpha\in\mathbf{C}$ and $p\in S(n,\beta)$, de…

For $n\ge 4$, consider the polynomial $p(z)=z^n+z$ and its rotations in the class $S_n$ of monic degree-$n$ polynomials with zeros in the closed unit disk. The paper proves that fo…

Let $S(n,0)$ be the set of monic complex polynomials of degree $n\ge 2$ whose zeros lie in the closed unit disk and which vanish at $0$. For $p\in S_n$, define … The paper has dete…

Let $S(n,0)$ be the set of monic complex polynomials of degree $n\ge 2$ whose zeros lie in the closed unit disk and which vanish at $0$. For $p\in S_n$, let $d(p)$ be the directed…

Zero distribution conjecture. The zeroes of the scaled polynomials $D_k^{(1)}(x/k)$ have the stated two-part asymptotic distribution.

Let $\beta$ be a complex number with $|\beta|\le1$, and let $S(\beta)$ denote the set of polynomials of degree at least $2$ with complex coefficients, all roots in the closed unit…

Let $\beta\in[0,1]$, let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$,…

Let $\beta\in[0,1]$, let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$,…

Let $\beta\in[0,1]$, let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$,…

For $n\ge2$, let $r_n(\beta)$ be the maximum of $d(P,\beta)$ over $P\in S(\beta)$ of degree $n$, and let $c_n$ denote the corresponding sharp quadratic-refinement coefficient in th…

Let $\beta$ be a real number in $[0,1]$. For a polynomial $P$ of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$,…

Let $\mathcal{A}$ be a unital C-algebra, let $n\in\mathbb{N}\setminus\{1\}$, and let … with $a_1,\dots,a_n\in\overline{\mathbb{D}^*(0,1)}$. Define … where the hatted factor is omit…

Let $\mathcal{A}$ be a unital commutative C-algebra whose group of invertible elements $G(\mathcal{A})$ is dense in $\mathcal{A}$. Let $n\in\mathbb{N}\setminus\{1\}$ and … where…

Schmeisser's conjecture. For every $\zeta\in\operatorname{conv}(V(F))$, the closed disk centered at $\zeta$ with radius $\sigma_\infty(F)$ contains a critical point of $F$.

Borcea's Toeplitz conjecture. The matrix $B$ has at least one eigenvalue $\lambda$ satisfying

Borcea's normal-matrix invertibility conjecture. If every eigenvalue of every $B_\ell$ lies outside the unit disk and

Borcea's univalence conjecture. For $p\geq1$, $F$ is not univalent in any closed disk centered at zero whose radius is larger than

Asymptotic Borcea variance conjecture. For all $n\geq2$ and $p\geq1$, there exists $C_{n,p}\geq1$ such that

Phelps–Rodrigues extremal conjecture. If $p>1$, equality in Borcea's variance inequality occurs if and only if

Borcea's variance conjecture. For every such $F$ and every $p\in[1,\infty)$,