36 problems

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 $n\ge3$, write $n=2s-1$ when $n$ is odd and $n=2s$ when $n$ is even, and let $r_n(t)$ be the polynomial defined in the paper. Write the Poincaré series in lowest terms as … whe…

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 $n\ge3$ be an integer, write $n=2s-1$ when $n$ is odd and $n=2s$ when $n$ is even, and let $p_n,q_n\in\mathbf Z[z,t]$ and $r_n\in\mathbf Z[t]$ be the polynomials defined in the…

Let $p$ be a prime, let $V_n^*$ be the space of binary $n$-ic forms over a two-dimensional vector space over $\mathbf{F}_p$, and let $V_n$ be its dual. Define the exponential sum o…

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

Let $n$ be a positive integer, and consider the stratum of binary $n$-forms having two roots of multiplicity two, with partition $2^21^{n-4}$. Generator conjecture. Based on comput…

Finite projective tropical invariants conjecture. For any set of binary forms, there exists a finite set of projective tropical invariants.

Let $f(x,y)$ be an integral binary quartic form of nonzero discriminant. Integer solutions $(x,y)$ to the quartic Thue equations … are considered up to the identification…

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 $n\geq 3$ and let $h$ be a fixed integer. Consider integral binary forms of degree $n$ that locally represent $h$, meaning that the equation $F(x,y)=h$ has a solution over…

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 $F(x,y)\in\mathbb{Z}[x,y]$ be an irreducible binary form of degree $n$ with $s+1$ nonzero coefficients, and let $h$ be a positive integer. Consider the number of integer soluti…

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 and $n\ge k$, let…

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…

Monic Shapiro conjecture. The addition map satisfies

Shapiro's conjecture. Every binary form of degree $d e$ can be written as the sum of $d$ $d$-th powers of forms of degree $e$.

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 $k$ be an algebraically closed field of characteristic $3$ or $7$, and let ${\cal C}_8$ denote the algebra of binary covariants of the octavic defined over $k$. Basson's genera…

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 $\mathcal{T}_{r,2,d}$ be the binary fradeco variety, and let $\mathcal{M}_r$ be a matrix whose properties (a), (b), and (c) are those specified in Theorem 3.1. All-ranks matrix…