20 problems

Bean's conjecture. The maximum is attained precisely when $F$, up to multiplication by a complex number, is equivalent under $\operatorname{GL}_2(\mathbb R)$ to $F_{n,n}$. In parti…

Binary-form area lower-bound conjecture. One has

Let $n\ge3$, and write the Poincaré series in lowest terms as … where $A_n,B_n\in\mathbf Z[t]$ are relatively prime and $A_n(0)=B_n(0)=1$. Irreducibility conjecture. The polynomial…

Let $n\ge3$ and let $I_n=\langle p_n,q_n\rangle\subset\mathbf Z[z,t]$, with $p_n$ and $q_n$ as defined in the paper. The intersection $I_n\cap\mathbf Z[t]$ is an ideal of…

Let $U$ be a two-dimensional vector space, let $a,b$ be positive integers, and write $S=\operatorname{Sym}(\operatorname{Sym}^b U)$. Let…

Bean's conjecture. The maximum value $M_n$ of $Q(F)$ over all such forms is attained precisely when $F$, up to multiplication by a complex number, is equivalent under…

Let $\lambda=[\lambda_1,\ldots,\lambda_n]$ be a partition, let $m_1(\lambda)$ denote the multiplicity of its parts equal to $1$, and let $\mathcal{F}_j(\lambda)$ be the family of d…

Let $S_d$ be the space of binary forms of degree $d$, let ${\rm Inv}_d$ be its ring of $SL_2$-invariants, and, for $d=2k$ with $k$ even, let $\mathscr{H}_{n,p}(F)$ denote the coeff…

Let $f=(f_1,f_2)$ be a basepoint-free pencil of binary quartics, viewed as a morphism $f\colon\mathbb{P}^1\to\mathbb{P}^1$. Let $(E,V,\phi,\xi)$ denote an algebraic quadruple, and…

Let $\operatorname{rk}^{\max}_k(kd,2)$ be the least number of $k$-th powers of degree-$d$ forms needed to represent every binary form of degree $kd$. Let $\ell_1$ and $\ell_2$ be n…

Let a binary form of degree $kd$ be expressed as a sum of $k$-th powers of binary forms of degree $d$, and let $\mathrm{rk}^{\max}_k(2,kd)$ denote the maximum of the resulting $k$-…

Let $d=2s+1\geq 7$. Given three general forms $f_1,f_2,f_3$ of degree $d$ in $R$, let $I=(f_1,f_2,f_3)$, let $\delta(I)$ be the ideal generated by the associated degree-$d-2$ forms…

Let $\mu\vdash d$ be a partition, let $\bar\mu$ denote the associated standard stratum, and let $\ell_{\bar\mu}$ and $\ell_{\mu'}$ denote the corresponding secant degeneracy length…

Let $\mu=(\mu_1\geq\mu_2\geq\dots\geq\mu_r)$ be a partition. It admits a strongly common radical solution if, for some integer $m$, a common radical solution exists for any choice…

Let $I$ be the ring of invariants of binary forms of degree $n$ under the action of $\mathrm{SL}(2,\mathbf{C})$. A homogeneous system of parameters (hsop) is an algebraically indep…

Let $p$ be a general binary form of even degree $2s$, and let $\alpha_j,\beta_j\in\mathbb{C}$ for $1\leq j\leq s+1$. Canonical representation conjecture. The form $p$ can be writte…

Assume $d=3$ and let $r$ be arbitrary, with $3$ not dividing $r$. Let $X\subseteq\mathbf P^3$ be the twisted cubic curve, let $I_X$ be its defining ideal, and let $J_{r,3}$ be the…

Assume that $r$ divides $d$. Let $X=X_{e,d}$ be the variety of binary $d$-ics that are perfect $\frac{d}{e}$-th powers of binary $e$-ics, and let $I_X$ be its defining ideal. Let…

Let $X^n$ be the space of homogeneous binary forms of degree $n$, let ${\mathcal R}^n$ be the collection of invariant rational functions on $X^n$ obtained by evaluating absolute cl…

Let $F$ be a separable homogeneous cubic binary form with integer coefficients, let $d$ be a fixed integer with $d\geq 1$, and let $l$ be a prime number. Cubic binary-form perfect-…