31 problems

Let $H_{n,m}(z)=p(z)+\overline{q(z)}$ be a harmonic polynomial with $\deg p=n$ and $\deg q=m<n$. The Wilmshurst refinement. For each $n,m$ with $m<n$, the maximum number of zeros o…

Let $H_{n,m}(z)=p(z)+\overline{q(z)}$ be a complex harmonic polynomial with $\deg p=n$ and $\deg q=m<n$. The Sheil-Small maximum valence conjecture. For each $n$, the maximum numbe…

Let $F(z)=p(z)+\overline{q(z)}$ be a harmonic polynomial with $p$ and $q$ of degrees $n$ and $m$, respectively, and let $N_F$ denote its number of zeros. Lee–Lerario–Lundberg conje…

Let $S(f)$ denote the stationary set of the solution of the wave equation in $\mathbb{R}^n$ with compactly supported initial data $f$, and suppose that stationary sets are containe…

Let $G(m,1,n)$ be the complex reflection group of $n\times n$ monomial matrices whose nonzero entries are $m$th roots of unity. Let the operator theorem refer to the type $B$ opera…

Let $n$ be a positive integer and let $a,b\in\mathbb{C}$ satisfy $|a|\neq 1$. Set $h(z,w)=z^n+aw^n+b$ and $h^{\ast}$ for the polynomial obtained by conjugating the coefficients and…

Let $m=\alpha n$ with $0<\alpha<1$, and consider $H_{n,m}(z)=p_n(z)+\overline{q_m(z)}$ in the Kac ensemble, where $q_m$ is an independent truncation to degree $m$ of the degree-$n$…

Let the coefficients of the harmonic polynomial model be independent and identically distributed random variables with mean zero, positive variance, and finite moments. The i.i.d.…

Let $p_n$ and $q_n$ be sampled independently from the Kac ensemble, and set … Let $N_H$ denote the number of zeros of $H$. The balanced harmonic Kac conjecture. As $n\to\infty$, th…

Let $H(z)=p_n(z)+\overline{q_m(z)}$, where $p_n$ and $q_m$ are sampled from the harmonic Kac model, and let $N_H$ denote the number of zeros of $H$. The Li–Wei asymptotic conjectur…

In the physical case, organize the complete set of generators in a lattice connected by nonzero symmetrized derivatives. The rim is the subset of generalized-harmonic generators at…

Let $d_j:\Omega_n\to\Omega_n$ be defined for $j\geq1$ by … and let $\delta_n\in\mathbb{C}[x_1,\dots,x_n]$ be the Vandermonde determinant. Let $SR_n$ be the superspace coinvariant a…

Let $N(n,m)$ denote the maximum valence of a harmonic polynomial $p(z)+\overline{q(z)}$, where $p$ and $q$ are complex polynomials of degrees $n>m>0$. Lundberg–Lerario–Lundberg con…

Let $p(z)$ and $q(z)$ be complex analytic polynomials of degrees $n$ and $m$, respectively, with $m<n$. The harmonic polynomial $p(z)-\overline{q(z)}$ has at most $n^2$ zeros. When…

Let $S_n$ act on polynomials in $n$ variables and on polynomial differential forms, and let $d_j$ denote the differential operators defined in the paper. Let $H$ be the space of al…

Let $H_{n,m}(z)=p(z)+q(\overline{z})$ be a random harmonic polynomial in the Weyl model, with $p$ and $q$ analytic complex polynomials of degrees $n$ and $m$, respectively. Let…

Let $H_{n,m}(z)=p(z)+q(\overline{z})$ be a harmonic polynomial, where $p$ and $q$ are analytic complex polynomials of degrees $n$ and $m$, respectively, with $n\geq m$. For fixed…

Let $p\in\mathbb{Z}^+$, define $p^*=p\mod 9$, and let $n_0(p)$ denote the first value of $n$ for which the two additional solutions in the paper's theorem occur. Threshold formula…

Let $p$ and $q$ be complex polynomials with degrees $n=\deg p$ and $m=\deg q$, and consider the harmonic polynomial $p+\overline q$. No-polynomial-formula conjecture. Given the deg…

Let $p$ and $q$ be complex polynomials with degrees $n=\deg p$ and $m=\deg q$, and let $p+\overline q$ be the associated harmonic polynomial. Write $k_\text{max}(n)$ for the quanti…

Let $h=p_n+\overline{q_m}$ be a complex harmonic polynomial, where $p_n$ and $q_m$ are analytic polynomials of degrees $n$ and $m$, respectively, and suppose that $m=n-2$. Conjectu…

Let $F(z)=p(z)+\overline{q(z)}$ be sampled from the truncated random harmonic-polynomial model, with $m=\alpha n$ and $0<\alpha<1$, and let $N_F$ be its number of zeros. Truncated-…

Let $S\subset\mathbb R^d$. Call $S$ an injectivity set for the spherical mean transform if vanishing of all spherical means centered on $S$ forces the underlying function or distri…

Let $S$ be a real-analytically ruled surface in $\mathbb R^3$ with no parallel generating lines, and suppose it is the common nodal set of a Paley–Wiener family of Laplace eigenfun…

Let $F(z)=p(z)+\overline{q(z)}$, where $p$ has degree $n$ and $q$ is drawn from the truncated version of the Kostlan ensemble described in the source. Assume that $m=\alpha n$ and…